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
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| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
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 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_tsub_eq_min | [11, 1] | [12, 70] | rw [tsub_eq_tsub_min _ b, tsub_tsub_cancel_of_le (min_le_left a _)] | α : Type u_1
inst✝³ : CanonicallyLinearOrderedAddCommMonoid α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
a b : α
⊢ a - (a - b) = min a b | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | classical
by_contra h
have hst : s ∩ t ⊂ s := by
use inter_subset_left
intro H
exact hxt.2 $ Set.mem_biUnion ⟨H.trans inter_subset_right,
fun H2 => h $ (H.trans inter_subset_right).antisymm H2⟩ hxs.1
refine' hxs.2 (Set.mem_biUnion hst _)
push_cast
exact K.inter_subset_convexHull hs ht ⟨hxs.1, hxt.1⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
⊢ s = t | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | by_contra h | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
⊢ s = t | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | have hst : s ∩ t ⊂ s := by
use inter_subset_left
intro H
exact hxt.2 $ Set.mem_biUnion ⟨H.trans inter_subset_right,
fun H2 => h $ (H.trans inter_subset_right).antisymm H2⟩ hxs.1 | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
⊢ False | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | refine' hxs.2 (Set.mem_biUnion hst _) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ False | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ x ∈ (convexHull 𝕜) ↑(s ∩ t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | push_cast | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ x ∈ (convexHull 𝕜) ↑(s ∩ t) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ x ∈ (convexHull 𝕜) (↑s ∩ ↑t) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | exact K.inter_subset_convexHull hs ht ⟨hxs.1, hxt.1⟩ | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
hst : s ∩ t ⊂ s
⊢ x ∈ (convexHull 𝕜) (↑s ∩ ↑t) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | use inter_subset_left | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
⊢ s ∩ t ⊂ s | case right
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
⊢ ¬s ⊆ s ∩ t |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | intro H | case right
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
⊢ ¬s ⊆ s ∩ t | case right
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
H : s ⊆ s ∩ t
⊢ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.disjoint_interiors | [33, 1] | [44, 55] | exact hxt.2 $ Set.mem_biUnion ⟨H.trans inter_subset_right,
fun H2 => h $ (H.trans inter_subset_right).antisymm H2⟩ hxs.1 | case right
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hs : s ∈ K
ht : t ∈ K
hxs : x ∈ combiInterior 𝕜 s
hxt : x ∈ combiInterior 𝕜 t
h : ¬s = t
H : s ⊆ s ∩ t
⊢ False | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.eq_singleton_of_singleton_mem_combiInterior | [50, 1] | [54, 28] | apply disjoint_interiors hs hx hxs | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : {x} ∈ K
hs : s ∈ K
hxs : x ∈ combiInterior 𝕜 s
⊢ s = {x} | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : {x} ∈ K
hs : s ∈ K
hxs : x ∈ combiInterior 𝕜 s
⊢ x ∈ combiInterior 𝕜 {x} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.eq_singleton_of_singleton_mem_combiInterior | [50, 1] | [54, 28] | rw [combiInterior_singleton] | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : {x} ∈ K
hs : s ∈ K
hxs : x ∈ combiInterior 𝕜 s
⊢ x ∈ combiInterior 𝕜 {x} | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : {x} ∈ K
hs : s ∈ K
hxs : x ∈ combiInterior 𝕜 s
⊢ x ∈ {x} |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.eq_singleton_of_singleton_mem_combiInterior | [50, 1] | [54, 28] | exact Set.mem_singleton x | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
hx : {x} ∈ K
hs : s ∈ K
hxs : x ∈ combiInterior 𝕜 s
⊢ x ∈ {x} | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.combiInteriors_cover | [56, 1] | [66, 56] | unfold space | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ ⋃ s ∈ K, combiInterior 𝕜 s = K.space | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ ⋃ s ∈ K, combiInterior 𝕜 s = ⋃ s ∈ K.faces, (convexHull 𝕜) ↑s |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/SimplicialComplex/Basic.lean | Geometry.SimplicialComplex.combiInteriors_cover | [56, 1] | [66, 56] | refine' (Set.iUnion₂_mono fun _ _ => _).antisymm (Set.iUnion₂_subset fun s hs => _) | 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
⊢ ⋃ s ∈ K, combiInterior 𝕜 s = ⋃ s ∈ K.faces, (convexHull 𝕜) ↑s | case refine'_1
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s t : Finset E
A : Set (Finset E)
m n : ℕ
x✝¹ : Finset E
x✝ : x✝¹ ∈ K
⊢ combiInterior 𝕜 x✝¹ ⊆ (convexHull 𝕜) ↑x✝¹
case refine'_2
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K K₁ K₂ : SimplicialComplex 𝕜 E
x y : E
s✝ t : Finset E
A : Set (Finset E)
m n : ℕ
s : Finset E
hs : s ∈ K
⊢ (convexHull 𝕜) ↑s ⊆ ⋃ i ∈ K, combiInterior 𝕜 i |
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