Theorem: RingSubgroupsBasis.hasBasis_nhds
File: Mathlib/Topology/Algebra/Nonarchimedean/Bases.lean

tactic: intro s
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
⊢ ∀ (t : Set A), t ∈ 𝓝 a ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ t
state_after: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ s ∈ 𝓝 a ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s

tactic: rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ s ∈ 𝓝 a ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s
state_after: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s

tactic: simp only [true_and]
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s
state_after: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, {b | b - a ∈ B i} ⊆ s

tactic: constructor
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, {b | b - a ∈ B i} ⊆ s
state_after: case mp
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) → ∃ i, {b | b - a ∈ B i} ⊆ s

case mpr
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i, {b | b - a ∈ B i} ⊆ s) → ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s

tactic: rintro ⟨-, ⟨i, rfl⟩, hi⟩
state_before: case mp
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) → ∃ i, {b | b - a ∈ B i} ⊆ s
state_after: case mp.intro.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ ∃ i, {b | b - a ∈ B i} ⊆ s

tactic: use i
state_before: case mp.intro.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ ∃ i, {b | b - a ∈ B i} ⊆ s
state_after: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} ⊆ s

tactic: suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i) by
  rw [h]
  assumption
state_before: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} ⊆ s
state_after: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)

tactic: simp only [image_add_left, neg_add_eq_sub]
state_before: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
state_after: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun x => x - a) ⁻¹' ↑(B i)

tactic: ext b
state_before: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun x => x - a) ⁻¹' ↑(B i)
state_after: case h.h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
b : A
⊢ b ∈ {b | b - a ∈ B i} ↔ b ∈ (fun x => x - a) ⁻¹' ↑(B i)

tactic: simp
state_before: case h.h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
b : A
⊢ b ∈ {b | b - a ∈ B i} ↔ b ∈ (fun x => x - a) ⁻¹' ↑(B i)
state_after: no goals

tactic: rw [h]
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ {b | b - a ∈ B i} ⊆ s
state_after: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ (fun y => a + y) '' ↑(B i) ⊆ s

tactic: assumption
state_before: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ (fun y => a + y) '' ↑(B i) ⊆ s
state_after: no goals

tactic: rintro ⟨i, hi⟩
state_before: case mpr
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i, {b | b - a ∈ B i} ⊆ s) → ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s
state_after: case mpr.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s

tactic: use B i
state_before: case mpr.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s
state_after: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis ∧ (fun y => a + y) '' ↑(B i) ⊆ s

tactic: constructor
state_before: case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis ∧ (fun y => a + y) '' ↑(B i) ⊆ s
state_after: case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis

case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ (fun y => a + y) '' ↑(B i) ⊆ s

tactic: use i
state_before: case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis
state_after: no goals

tactic: rw [image_subset_iff]
state_before: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ (fun y => a + y) '' ↑(B i) ⊆ s
state_after: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ⊆ (fun y => a + y) ⁻¹' s

tactic: rintro b b_in
state_before: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ⊆ (fun y => a + y) ⁻¹' s
state_after: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ b ∈ (fun y => a + y) ⁻¹' s

tactic: apply hi
state_before: case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ b ∈ (fun y => a + y) ⁻¹' s
state_after: case h.right.a
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ (fun y => a + y) b ∈ {b | b - a ∈ B i}

tactic: simpa using b_in
state_before: case h.right.a
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ (fun y => a + y) b ∈ {b | b - a ∈ B i}
state_after: no goals