Theorem: Subsemigroup.mem_iSup_of_directed
File: Mathlib/Algebra/Group/Subsemigroup/Membership.lean

tactic: refine ⟨?_, fun ⟨i, hi⟩ ↦ le_iSup S i hi⟩
state_before: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i
state_after: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i

tactic: suffices x ∈ closure (⋃ i, (S i : Set M)) → ∃ i, x ∈ S i by
  simpa only [closure_iUnion, closure_eq (S _)] using this
state_before: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i
state_after: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
⊢ x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i

tactic: refine fun hx ↦ closure_induction hx (fun y hy ↦ mem_iUnion.mp hy) ?_
state_before: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
⊢ x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i
state_after: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
hx : x ∈ closure (⋃ i, ↑(S i))
⊢ ∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i

tactic: rintro x y ⟨i, hi⟩ ⟨j, hj⟩
state_before: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
hx : x ∈ closure (⋃ i, ↑(S i))
⊢ ∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i
state_after: case intro.intro
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x✝ : M
hx : x✝ ∈ closure (⋃ i, ↑(S i))
x y : M
i : ι
hi : x ∈ S i
j : ι
hj : y ∈ S j
⊢ ∃ i, x * y ∈ S i

tactic: rcases hS i j with ⟨k, hki, hkj⟩
state_before: case intro.intro
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x✝ : M
hx : x✝ ∈ closure (⋃ i, ↑(S i))
x y : M
i : ι
hi : x ∈ S i
j : ι
hj : y ∈ S j
⊢ ∃ i, x * y ∈ S i
state_after: case intro.intro.intro.intro
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x✝ : M
hx : x✝ ∈ closure (⋃ i, ↑(S i))
x y : M
i : ι
hi : x ∈ S i
j : ι
hj : y ∈ S j
k : ι
hki : S i ≤ S k
hkj : S j ≤ S k
⊢ ∃ i, x * y ∈ S i

tactic: exact ⟨k, (S k).mul_mem (hki hi) (hkj hj)⟩
state_before: case intro.intro.intro.intro
ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x✝ : M
hx : x✝ ∈ closure (⋃ i, ↑(S i))
x y : M
i : ι
hi : x ∈ S i
j : ι
hj : y ∈ S j
k : ι
hki : S i ≤ S k
hkj : S j ≤ S k
⊢ ∃ i, x * y ∈ S i
state_after: no goals

tactic: simpa only [closure_iUnion, closure_eq (S _)] using this
state_before: ι : Sort u_1
M : Type u_2
A : Type u_3
B : Type u_4
inst✝ : Mul M
S : ι → Subsemigroup M
hS : Directed (fun x x_1 => x ≤ x_1) S
x : M
this : x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i
⊢ x ∈ ⨆ i, S i → ∃ i, x ∈ S i
state_after: no goals