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train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	letI : Fintype (stabilizer α t) := fintypeStabilizerOfMulStab ht	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs ⊢ s.mulStab.card ∣ t.mulStab.card	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card ∣ t.mulStab.card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	convert Subgroup.card_dvd_of_le h using 1	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card ∣ t.mulStab.card	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = Nat.card ↥(stabilizer α s) case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	simp [-mem_stabilizer_iff]	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = Nat.card ↥(stabilizer α s) case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = Fintype.card ↥(stabilizer α s) case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	change _ = (s.mulStab.attach.map ⟨Subtype.map id fun _ ↦ (mem_mulStab hs).1, Subtype.map_injective _ injective_id⟩).card	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = Fintype.card ↥(stabilizer α s) case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = (map { toFun := Subtype.map id ⋯, inj' := ⋯ } s.mulStab.attach).card case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	simp	case h.e'_3 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ s.mulStab.card = (map { toFun := Subtype.map id ⋯, inj' := ⋯ } s.mulStab.attach).card case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	simp [-mem_stabilizer_iff]	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Nat.card ↥(stabilizer α t)	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Fintype.card ↥(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	change _ = (t.mulStab.attach.map ⟨Subtype.map id fun _ ↦ (mem_mulStab ht).1, Subtype.map_injective _ injective_id⟩).card	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = Fintype.card ↥(stabilizer α t)	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = (map { toFun := Subtype.map id ⋯, inj' := ⋯ } t.mulStab.attach).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	simp	case h.e'_4 ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t : Finset α a : α hs : s.Nonempty h : stabilizer α s ≤ stabilizer α t ht : t.Nonempty this✝ : Fintype ↥(stabilizer α s) := Finset.fintypeStabilizerOfMulStab hs this : Fintype ↥(stabilizer α t) := Finset.fintypeStabilizerOfMulStab ht ⊢ t.mulStab.card = (map { toFun := Subtype.map id ⋯, inj' := ⋯ } t.mulStab.attach).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_dvd_card_mulStab	[265, 1]	[281, 7]	simp	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s : Finset α a : α hs : s.Nonempty h : s.mulStab ⊆ ∅.mulStab ⊢ s.mulStab.card ∣ ∅.mulStab.card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	obtain rfl \| ht := t.eq_empty_or_nonempty	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α ⊢ ∅.mulStab.card * (image QuotientGroup.mk s).card = (s * ∅.mulStab).card case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	have := QuotientGroup.preimageMkEquivSubgroupProdSet _ (s +ˢ stabilizer α t)	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	have that : ↥(stabilizer α t) = ↥t.mulStab := by rw [← SetLike.coe_sort_coe, ← coe_mulStab ht, Finset.coe_sort_coe]	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	have temp := this.trans ((Equiv.cast that).prodCongr (Equiv.refl _))	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } temp : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ { x // x ∈ t.mulStab } × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	rw [preimage_image_quotientMk_stabilizer_eq_mul_mulStab ht] at temp	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } temp : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ { x // x ∈ t.mulStab } × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } temp : ↑(↑s * ↑t.mulStab) ≃ { x // x ∈ t.mulStab } × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	simpa only [coe_sort_coe, ← coe_mul, Fintype.card_prod, Fintype.card_coe, Fintype.card_ofFinset, toFinset_coe, mem_image, Set.mem_image, mem_coe, forall_const, eq_comm] using Fintype.card_congr temp	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) that : ↥(stabilizer α t) = { x // x ∈ t.mulStab } temp : ↑(↑s * ↑t.mulStab) ≃ { x // x ∈ t.mulStab } × ↑(QuotientGroup.mk '' ↑s) ⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	simp	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α ⊢ ∅.mulStab.card * (image QuotientGroup.mk s).card = (s * ∅.mulStab).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe'	[283, 1]	[296, 34]	rw [← SetLike.coe_sort_coe, ← coe_mulStab ht, Finset.coe_sort_coe]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s) ⊢ ↥(stabilizer α t) = { x // x ∈ t.mulStab }	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	obtain rfl \| hs := s.eq_empty_or_nonempty	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t✝ : Finset α a : α t : Finset α ⊢ (∅ * t).card = (∅ * t).mulStab.card * (image QuotientGroup.mk (∅ * t)).card case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	obtain rfl \| ht := t.eq_empty_or_nonempty	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inr.inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α hs : s.Nonempty ⊢ (s * ∅).card = (s * ∅).mulStab.card * (image QuotientGroup.mk (s * ∅)).card case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	have := QuotientGroup.preimageMkEquivSubgroupProdSet _ $ ↑(s * t) +ˢ stabilizer α (s * t)	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	have that : ↥(stabilizer α (s * t)) = ↥(s * t).mulStab := by rw [← SetLike.coe_sort_coe, ← coe_mulStab (hs.mul ht), Finset.coe_sort_coe]	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	have temp := this.trans $ (Equiv.cast that).prodCongr (Equiv.refl _)	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } temp : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	rw [preimage_image_quotientMk_mulStabilizer] at temp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } temp : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } temp : ↑↑(s * t) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	simpa [-coe_mul] using Fintype.card_congr temp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) that : ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab } temp : ↑↑(s * t) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ (s * t).card = (s * t).mulStab.card * (image QuotientGroup.mk (s * t)).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	simp	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t✝ : Finset α a : α t : Finset α ⊢ (∅ * t).card = (∅ * t).mulStab.card * (image QuotientGroup.mk (∅ * t)).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	simp	case inr.inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α hs : s.Nonempty ⊢ (s * ∅).card = (s * ∅).mulStab.card * (image QuotientGroup.mk (s * ∅)).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mul_card_eq_mulStab_card_mul_coe	[298, 1]	[310, 49]	rw [← SetLike.coe_sort_coe, ← coe_mulStab (hs.mul ht), Finset.coe_sort_coe]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑(s * t))) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑(s * t)) ⊢ ↥(stabilizer α (s * t)) = { x // x ∈ (s * t).mulStab }	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	obtain rfl \| hs := s.eq_empty_or_nonempty	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t✝ : Finset α a : α t : Finset α ⊢ (∅ * t).mulStab.card * (image QuotientGroup.mk ∅ * image QuotientGroup.mk t).card = (∅ * t).card case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	obtain rfl \| ht := t.eq_empty_or_nonempty	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α hs : s.Nonempty ⊢ (s * ∅).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk ∅).card = (s * ∅).card case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have := QuotientGroup.preimageMkEquivSubgroupProdSet (stabilizer α (s * t)) ((s +ˢ stabilizer α (s * t)) * (t +ˢ stabilizer α (s * t)))	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have image_coe_mul : ((s * t) +ˢ stabilizer α (s * t)) = (s +ˢ stabilizer α (s * t)) * (t +ˢ stabilizer α (s * t)) := Set.image_mul (QuotientGroup.mk' (stabilizer α (s * t)) : α →* α ⧸ stabilizer α (s * t))	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	rw [← image_coe_mul, ← coe_mul, preimage_image_quotientMk_mulStabilizer, coe_mul, image_coe_mul] at this	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have that : (stabilizer α (s * t) × ↥((s +ˢ stabilizer α (s * t)) * (t +ˢ stabilizer α (s * t)))) = ((s * t).mulStab × ↥((s +ˢ stabilizer α (s * t)) * (t +ˢ stabilizer α (s * t)))) := by rw [← SetLike.coe_sort_coe, ← coe_mulStab (hs.mul ht), Finset.coe_sort_coe]	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have temp := this.trans (Equiv.cast that)	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : ↑(↑s * ↑t) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	replace temp := Fintype.card_congr temp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : ↑(↑s * ↑t) ≃ { x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑(↑s * ↑t) = Fintype.card ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	simp_rw [← Finset.coe_mul s t] at temp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑(↑s * ↑t) = Fintype.card ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = Fintype.card ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	simp only [Fintype.card_prod, Fintype.card_coe] at temp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = Fintype.card ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have h1 : Fintype.card ((s * t : Finset α) : Set α) = Fintype.card (s * t) := by congr	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) h1 : Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t } ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	have h2 : (s +ˢ stabilizer α (s * t)) * (t +ˢ stabilizer α (s * t)) = ↑((s +ₛ stabilizer α (s * t)) * (t +ₛ stabilizer α (s * t))) := by simp	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) h1 : Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t } ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) h1 : Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t } h2 : QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t = ↑(image QuotientGroup.mk s * image QuotientGroup.mk t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	sorry	case inr.inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) h1 : Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t } h2 : QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t = ↑(image QuotientGroup.mk s * image QuotientGroup.mk t) ⊢ (s * t).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk t).card = (s * t).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	simp	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s t✝ : Finset α a : α t : Finset α ⊢ (∅ * t).mulStab.card * (image QuotientGroup.mk ∅ * image QuotientGroup.mk t).card = (∅ * t).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	simp	case inr.inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Finset α hs : s.Nonempty ⊢ (s * ∅).mulStab.card * (image QuotientGroup.mk s * image QuotientGroup.mk ∅).card = (s * ∅).card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	rw [← SetLike.coe_sort_coe, ← coe_mulStab (hs.mul ht), Finset.coe_sort_coe]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t ⊢ (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t))	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	congr	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) ⊢ Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t }	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.card_mulStab_mul_card_image_coe	[312, 1]	[339, 8]	simp	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s t : Finset α hs : s.Nonempty ht : t.Nonempty this : ↑(↑s * ↑t) ≃ ↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) image_coe_mul : QuotientGroup.mk '' (↑s * ↑t) = QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t that : (↥(stabilizer α (s * t)) × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) = ({ x // x ∈ (s * t).mulStab } × ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t)) temp : Fintype.card ↑↑(s * t) = (s * t).mulStab.card * Fintype.card ↑(QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t) h1 : Fintype.card ↑↑(s * t) = Fintype.card { x // x ∈ s * t } ⊢ QuotientGroup.mk '' ↑s * QuotientGroup.mk '' ↑t = ↑(image QuotientGroup.mk s * image QuotientGroup.mk t)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	have := s.subgroup_mul_card_eq_mul t	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab h : ↑t * ↑s = ↑t ⊢ Nat.card ↥s * (image QuotientGroup.mk t).card = t.card	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab h : ↑t * ↑s = ↑t this : Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' ↑t) = Nat.card ↑(↑t * ↑s) ⊢ Nat.card ↥s * (image QuotientGroup.mk t).card = t.card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	rw [h] at this	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab h : ↑t * ↑s = ↑t this : Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' ↑t) = Nat.card ↑(↑t * ↑s) ⊢ Nat.card ↥s * (image QuotientGroup.mk t).card = t.card	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab h : ↑t * ↑s = ↑t this : Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' ↑t) = Nat.card ↑↑t ⊢ Nat.card ↥s * (image QuotientGroup.mk t).card = t.card
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	simpa	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab h : ↑t * ↑s = ↑t this : Nat.card ↥s * Nat.card ↑(QuotientGroup.mk '' ↑t) = Nat.card ↑↑t ⊢ Nat.card ↥s * (image QuotientGroup.mk t).card = t.card	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	apply Set.Subset.antisymm (Set.Subset.trans (Set.mul_subset_mul_left hst) _)	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ ↑t * ↑s = ↑t	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ ↑t ⊆ ↑t * ↑s ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ ↑t * ↑t.mulStab ⊆ ↑t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	intro x	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ ↑t ⊆ ↑t * ↑s	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab x : α ⊢ x ∈ ↑t → x ∈ ↑t * ↑s
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	rw [Set.mem_mul]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab x : α ⊢ x ∈ ↑t → x ∈ ↑t * ↑s	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab x : α ⊢ x ∈ ↑t → ∃ x_1 ∈ ↑t, ∃ y ∈ ↑s, x_1 * y = x
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	aesop	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab x : α ⊢ x ∈ ↑t → ∃ x_1 ∈ ↑t, ∃ y ∈ ↑s, x_1 * y = x	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.subgroup_mul_card_eq_mul_of_mul_stab_subset	[348, 1]	[359, 8]	rw [← coe_mul, mul_mulStab]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ ↑t * ↑t.mulStab ⊆ ↑t	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	obtain rfl \| ht := t.eq_empty_or_nonempty	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ⊢ image QuotientGroup.mk t.mulStab = (image QuotientGroup.mk t).mulStab	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Subgroup α hst : ↑s ⊆ ↑∅.mulStab ⊢ image QuotientGroup.mk ∅.mulStab = (image QuotientGroup.mk ∅).mulStab case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty ⊢ image QuotientGroup.mk t.mulStab = (image QuotientGroup.mk t).mulStab
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	have hti : (image (QuotientGroup.mk (s := s)) t).Nonempty := by aesop	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty ⊢ image QuotientGroup.mk t.mulStab = (image QuotientGroup.mk t).mulStab	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty ⊢ image QuotientGroup.mk t.mulStab = (image QuotientGroup.mk t).mulStab
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	ext x	case inr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty ⊢ image QuotientGroup.mk t.mulStab = (image QuotientGroup.mk t).mulStab	case inr.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ x ∈ image QuotientGroup.mk t.mulStab ↔ x ∈ (image QuotientGroup.mk t).mulStab
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simp only [mem_image, image_nonempty, mem_mulStab hti]	case inr.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ x ∈ image QuotientGroup.mk t.mulStab ↔ x ∈ (image QuotientGroup.mk t).mulStab	case inr.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ (∃ a ∈ t.mulStab, ↑a = x) ↔ x • image QuotientGroup.mk t = image QuotientGroup.mk t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	constructor	case inr.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ (∃ a ∈ t.mulStab, ↑a = x) ↔ x • image QuotientGroup.mk t = image QuotientGroup.mk t	case inr.a.mp ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ (∃ a ∈ t.mulStab, ↑a = x) → x • image QuotientGroup.mk t = image QuotientGroup.mk t case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ x • image QuotientGroup.mk t = image QuotientGroup.mk t → ∃ a ∈ t.mulStab, ↑a = x
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simp	case inl ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t : Finset α a : α s : Subgroup α hst : ↑s ⊆ ↑∅.mulStab ⊢ image QuotientGroup.mk ∅.mulStab = (image QuotientGroup.mk ∅).mulStab	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	aesop	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty ⊢ (image QuotientGroup.mk t).Nonempty	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rintro ⟨a, hax⟩	case inr.a.mp ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ (∃ a ∈ t.mulStab, ↑a = x) → x • image QuotientGroup.mk t = image QuotientGroup.mk t	case inr.a.mp.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x ⊢ x • image QuotientGroup.mk t = image QuotientGroup.mk t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← hax.2]	case inr.a.mp.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x ⊢ x • image QuotientGroup.mk t = image QuotientGroup.mk t	case inr.a.mp.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x ⊢ ↑a • image QuotientGroup.mk t = image QuotientGroup.mk t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	ext z	case inr.a.mp.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x ⊢ ↑a • image QuotientGroup.mk t = image QuotientGroup.mk t	case inr.a.mp.intro.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ z ∈ ↑a • image QuotientGroup.mk t ↔ z ∈ image QuotientGroup.mk t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simp only [mem_smul_finset, mem_image, smul_eq_mul, exists_exists_and_eq_and]	case inr.a.mp.intro.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ z ∈ ↑a • image QuotientGroup.mk t ↔ z ∈ image QuotientGroup.mk t	case inr.a.mp.intro.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a_1 ∈ t, ↑a * ↑a_1 = z) ↔ ∃ a ∈ t, ↑a = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	constructor	case inr.a.mp.intro.a ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a_1 ∈ t, ↑a * ↑a_1 = z) ↔ ∃ a ∈ t, ↑a = z	case inr.a.mp.intro.a.mp ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a_1 ∈ t, ↑a * ↑a_1 = z) → ∃ a ∈ t, ↑a = z case inr.a.mp.intro.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a ∈ t, ↑a = z) → ∃ a_2 ∈ t, ↑a * ↑a_2 = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rintro ⟨b, hbt, hbaz⟩	case inr.a.mp.intro.a.mp ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a_1 ∈ t, ↑a * ↑a_1 = z) → ∃ a ∈ t, ↑a = z	case inr.a.mp.intro.a.mp.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ ∃ a ∈ t, ↑a = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	use (b * a)	case inr.a.mp.intro.a.mp.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ ∃ a ∈ t, ↑a = z	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ b * a ∈ t ∧ ↑(b * a) = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← mul_mulStab t]	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ b * a ∈ t ∧ ↑(b * a) = z	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ b * a ∈ t * t.mulStab ∧ ↑(b * a) = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	refine ⟨mul_mem_mul hbt hax.1, ?_⟩	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ b * a ∈ t * t.mulStab ∧ ↑(b * a) = z	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ ↑(b * a) = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← hbaz, QuotientGroup.mk_mul, mul_comm]	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbaz : ↑a * ↑b = z ⊢ ↑(b * a) = z	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rintro ⟨b, hbt, hbz⟩	case inr.a.mp.intro.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s ⊢ (∃ a ∈ t, ↑a = z) → ∃ a_2 ∈ t, ↑a * ↑a_2 = z	case inr.a.mp.intro.a.mpr.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ ∃ a_1 ∈ t, ↑a * ↑a_1 = z
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← hbz, ← mul_mulStab t, mul_comm]	case inr.a.mp.intro.a.mpr.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ ∃ a_1 ∈ t, ↑a * ↑a_1 = z	case inr.a.mp.intro.a.mpr.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ ∃ a_1 ∈ t.mulStab * t, ↑a * ↑a_1 = ↑b
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	use (a⁻¹ * b)	case inr.a.mp.intro.a.mpr.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ ∃ a_1 ∈ t.mulStab * t, ↑a * ↑a_1 = ↑b	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ * b ∈ t.mulStab * t ∧ ↑a * ↑(a⁻¹ * b) = ↑b
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	refine ⟨mul_mem_mul ?_ hbt, by simp⟩	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ * b ∈ t.mulStab * t ∧ ↑a * ↑(a⁻¹ * b) = ↑b	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ ∈ t.mulStab
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← mem_coe, coe_mulStab ht]	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ ∈ t.mulStab	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ ∈ ↑(stabilizer α t)
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	aesop	case h ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ a⁻¹ ∈ ↑(stabilizer α t)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simp	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s a : α hax : a ∈ t.mulStab ∧ ↑a = x z : α ⧸ s b : α hbt : b ∈ t hbz : ↑b = z ⊢ ↑a * ↑(a⁻¹ * b) = ↑b	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	intro hx	case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s ⊢ x • image QuotientGroup.mk t = image QuotientGroup.mk t → ∃ a ∈ t.mulStab, ↑a = x	case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t ⊢ ∃ a ∈ t.mulStab, ↑a = x
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	have : s ≤ stabilizer α t := by aesop	case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t ⊢ ∃ a ∈ t.mulStab, ↑a = x	case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t ⊢ ∃ a ∈ t.mulStab, ↑a = x
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	obtain ⟨y, hyx⟩ := Quotient.exists_rep x	case inr.a.mpr ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t ⊢ ∃ a ∈ t.mulStab, ↑a = x	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x ⊢ ∃ a ∈ t.mulStab, ↑a = x
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	refine ⟨y, (mem_mulStab_iff_subset_smul_finset ht).mpr ?_, by simpa⟩	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x ⊢ ∃ a ∈ t.mulStab, ↑a = x	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x ⊢ t ⊆ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	intros z hzt	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x ⊢ t ⊆ y • t	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	replace hx : image QuotientGroup.mk (y • t) = image (QuotientGroup.mk (s := s)) t := by rw [← hx, ← hyx] exact image_smul_comm QuotientGroup.mk y t (congrFun rfl)	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t ⊢ z ∈ y • t	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	have hyz : QuotientGroup.mk z ∈ image (QuotientGroup.mk (s := s)) (y • t) := by aesop	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t ⊢ z ∈ y • t	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t hyz : ↑z ∈ image QuotientGroup.mk (y • t) ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simp only [QuotientGroup.mk_mul, mem_image] at hyz	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t hyz : ↑z ∈ image QuotientGroup.mk (y • t) ⊢ z ∈ y • t	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t hyz : ∃ a ∈ y • t, ↑a = ↑z ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	obtain ⟨a, ha, hayz⟩ := hyz	case inr.a.mpr.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t hyz : ∃ a ∈ y • t, ↑a = ↑z ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t a : α ha : a ∈ y • t hayz : ↑a = ↑z ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	obtain ⟨b, hbt, haby⟩ := mem_smul_finset.mp ha	case inr.a.mpr.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t a : α ha : a ∈ y • t hayz : ↑a = ↑z ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t a : α ha : a ∈ y • t hayz : ↑a = ↑z b : α hbt : b ∈ t haby : y • b = a ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	subst a	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a✝ : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t a : α ha : a ∈ y • t hayz : ↑a = ↑z b : α hbt : b ∈ t haby : y • b = a ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : ↑(y • b) = ↑z ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [QuotientGroup.eq, smul_eq_mul] at hayz	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : ↑(y • b) = ↑z ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	replace : ∃ c ∈ mulStab t, (y • b)⁻¹ * z = c := by aesop	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s this : ∃ c ∈ t.mulStab, (y • b)⁻¹ * z = c ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	obtain ⟨c, hct, hcbyz⟩ := this	case inr.a.mpr.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s this : ∃ c ∈ t.mulStab, (y • b)⁻¹ * z = c ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : (y • b)⁻¹ * z = c ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [inv_mul_eq_iff_eq_mul] at hcbyz	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : (y • b)⁻¹ * z = c ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : z = y • b * c ⊢ z ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [hcbyz, smul_mul_assoc, mul_comm, ← smul_eq_mul]	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : z = y • b * c ⊢ z ∈ y • t	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : z = y • b * c ⊢ y • c • b ∈ y • t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	exact smul_mem_smul_finset ((mem_mulStab' ht).mp hct hbt)	case inr.a.mpr.intro.intro.intro.intro.intro.intro.intro ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s c : α hct : c ∈ t.mulStab hcbyz : z = y • b * c ⊢ y • c • b ∈ y • t	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	aesop	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t ⊢ s ≤ stabilizer α t	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	simpa	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x ⊢ ↑y = x	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	rw [← hx, ← hyx]	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t ⊢ image QuotientGroup.mk (y • t) = image QuotientGroup.mk t	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t ⊢ image QuotientGroup.mk (y • t) = ⟦y⟧ • image QuotientGroup.mk t
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	exact image_smul_comm QuotientGroup.mk y t (congrFun rfl)	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t ⊢ image QuotientGroup.mk (y • t) = ⟦y⟧ • image QuotientGroup.mk t	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	aesop	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t ⊢ ↑z ∈ image QuotientGroup.mk (y • t)	no goals
https://github.com/YaelDillies/LeanCamCombi.git	034199694e3b91536d03bc4a8b0cdbd659cdf50f	LeanCamCombi/Kneser/MulStab.lean	Finset.mulStab_quotient_commute_subgroup	[361, 1]	[404, 62]	aesop	ι : Type u_1 α : Type u_2 inst✝¹ : CommGroup α inst✝ : DecidableEq α s✝ t✝ : Finset α a : α s : Subgroup α t : Finset α hst : ↑s ⊆ ↑t.mulStab ht : t.Nonempty hti : (image QuotientGroup.mk t).Nonempty x : α ⧸ s this : s ≤ stabilizer α t y : α hyx : ⟦y⟧ = x z : α hzt : z ∈ t hx : image QuotientGroup.mk (y • t) = image QuotientGroup.mk t b : α hbt : b ∈ t ha : y • b ∈ y • t hayz : (y * b)⁻¹ * z ∈ s ⊢ ∃ c ∈ t.mulStab, (y • b)⁻¹ * z = c	no goals

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