[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
f : G →* H
emb : OpenEmbedding ↑f
U : Set H
hU : U ∈ nhds 1
⊢ ↑f ⁻¹' U ∈ nhds 1
[PROOFSTEP]
apply emb.continuous.tendsto
[GOAL]
case a
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
f : G →* H
emb : OpenEmbedding ↑f
U : Set H
hU : U ∈ nhds 1
⊢ U ∈ nhds (↑f 1)
[PROOFSTEP]
rwa [f.map_one]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
hU : U ∈ nhds 1
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
erw [nhds_prod_eq, Filter.mem_prod_iff] at hU 
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
hU : ∃ t₁, t₁ ∈ nhds 1 ∧ ∃ t₂, t₂ ∈ nhds 1 ∧ t₁ ×ˢ t₂ ⊆ U
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
rcases hU with ⟨U₁, hU₁, U₂, hU₂, h⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
cases' is_nonarchimedean _ hU₁ with V hV
[GOAL]
case intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
cases' is_nonarchimedean _ hU₂ with W hW
[GOAL]
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
W : OpenSubgroup K
hW : ↑W ⊆ U₂
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
use V
[GOAL]
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
W : OpenSubgroup K
hW : ↑W ⊆ U₂
⊢ ∃ W, ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
use W
[GOAL]
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
W : OpenSubgroup K
hW : ↑W ⊆ U₂
⊢ ↑V ×ˢ ↑W ⊆ U
[PROOFSTEP]
rw [Set.prod_subset_iff]
[GOAL]
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
W : OpenSubgroup K
hW : ↑W ⊆ U₂
⊢ ∀ (x : G), x ∈ ↑V → ∀ (y : K), y ∈ ↑W → (x, y) ∈ U
[PROOFSTEP]
intro x hX y hY
[GOAL]
case h
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × K)
U₁ : Set G
hU₁ : U₁ ∈ nhds 1
U₂ : Set K
hU₂ : U₂ ∈ nhds 1
h : U₁ ×ˢ U₂ ⊆ U
V : OpenSubgroup G
hV : ↑V ⊆ U₁
W : OpenSubgroup K
hW : ↑W ⊆ U₂
x : G
hX : x ∈ ↑V
y : K
hY : y ∈ ↑W
⊢ (x, y) ∈ U
[PROOFSTEP]
exact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY)
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : OpenSubgroup G
h : ↑V ×ˢ ↑W ⊆ U
⊢ ↑(V ⊓ W) ×ˢ ↑(V ⊓ W) ⊆ U
[PROOFSTEP]
refine' Set.Subset.trans (Set.prod_mono _ _) ‹_›
[GOAL]
case refine'_1
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : OpenSubgroup G
h : ↑V ×ˢ ↑W ⊆ U
⊢ ↑(V ⊓ W) ⊆ ↑V
[PROOFSTEP]
simp
[GOAL]
case refine'_2
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : NonarchimedeanGroup G
H : Type u_2
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : TopologicalGroup H
K : Type u_3
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : NonarchimedeanGroup K
U : Set (G × G)
hU : U ∈ nhds 1
V W : OpenSubgroup G
h : ↑V ×ˢ ↑W ⊆ U
⊢ ↑(V ⊓ W) ⊆ ↑W
[PROOFSTEP]
simp
[GOAL]
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U : OpenAddSubgroup R
⊢ ∃ V, ↑V * ↑V ⊆ ↑U
[PROOFSTEP]
let ⟨V, H⟩ :=
  prod_self_subset
    (IsOpen.mem_nhds (IsOpen.preimage continuous_mul U.isOpen)
      (by simpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero, mul_zero] using U.zero_mem))
[GOAL]
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U : OpenAddSubgroup R
⊢ 0 ∈ (fun p => p.fst * p.snd) ⁻¹' ↑U
[PROOFSTEP]
simpa only [Set.mem_preimage, SetLike.mem_coe, Prod.snd_zero, mul_zero] using U.zero_mem
[GOAL]
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
⊢ ∃ V, ↑V * ↑V ⊆ ↑U
[PROOFSTEP]
use V
[GOAL]
case h
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
⊢ ↑V * ↑V ⊆ ↑U
[PROOFSTEP]
rintro v ⟨a, b, ha, hb, hv⟩
[GOAL]
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1 => x * x_1) a b = v
⊢ v ∈ ↑U
[PROOFSTEP]
have hy := H (Set.mk_mem_prod ha hb)
[GOAL]
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1 => x * x_1) a b = v
hy : (a, b) ∈ (fun p => p.fst * p.snd) ⁻¹' ↑U
⊢ v ∈ ↑U
[PROOFSTEP]
simp only [Set.mem_preimage, SetLike.mem_coe, hv] at hy 
[GOAL]
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1 => x * x_1) a b = v
hy : v ∈ U
⊢ v ∈ ↑U
[PROOFSTEP]
rw [SetLike.mem_coe]
[GOAL]
case h.intro.intro.intro.intro
R : Type u_1
S : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : TopologicalSpace R
inst✝³ : NonarchimedeanRing R
inst✝² : Ring S
inst✝¹ : TopologicalSpace S
inst✝ : NonarchimedeanRing S
U V : OpenAddSubgroup R
H : ↑V ×ˢ ↑V ⊆ (fun p => p.fst * p.snd) ⁻¹' ↑U
v a b : R
ha : a ∈ ↑V
hb : b ∈ ↑V
hv : (fun x x_1 => x * x_1) a b = v
hy : v ∈ U
⊢ v ∈ U
[PROOFSTEP]
exact hy
