Theorem: Subalgebra.op_adjoin
File: Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean

Proof steps:
state_before: ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op = Algebra.adjoin R (MulOpposite.unop ⁻¹' s)
state_after: case a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op.toSubsemiring = (Algebra.adjoin R (MulOpposite.unop ⁻¹' s)).toSubsemiring
tactic: apply toSubsemiring_injective
state_before: case a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ (Algebra.adjoin R s).op.toSubsemiring = (Algebra.adjoin R (MulOpposite.unop ⁻¹' s)).toSubsemiring
state_after: case a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ Subsemiring.closure (MulOpposite.unop ⁻¹' Set.range ⇑(algebraMap R A) ∪ MulOpposite.unop ⁻¹' s) =
    Subsemiring.closure (Set.range ⇑(algebraMap R Aᵐᵒᵖ) ∪ MulOpposite.unop ⁻¹' s)
tactic: simp_rw [Algebra.adjoin, op_toSubsemiring, Subsemiring.op_closure, Set.preimage_union]
state_before: case a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ Subsemiring.closure (MulOpposite.unop ⁻¹' Set.range ⇑(algebraMap R A) ∪ MulOpposite.unop ⁻¹' s) =
    Subsemiring.closure (Set.range ⇑(algebraMap R Aᵐᵒᵖ) ∪ MulOpposite.unop ⁻¹' s)
state_after: case a.e_s.e_a.h
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
⊢ x ∈ MulOpposite.unop ⁻¹' Set.range ⇑(algebraMap R A) ↔ x ∈ Set.range ⇑(algebraMap R Aᵐᵒᵖ)
tactic: congr with x
state_before: case a.e_s.e_a.h
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
⊢ x ∈ MulOpposite.unop ⁻¹' Set.range ⇑(algebraMap R A) ↔ x ∈ Set.range ⇑(algebraMap R Aᵐᵒᵖ)
state_after: case a.e_s.e_a.h
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
⊢ (∃ y, (algebraMap R A) y = MulOpposite.unop x) ↔ ∃ y, MulOpposite.op ((algebraMap R A) y) = x
tactic: simp_rw [Set.mem_preimage, Set.mem_range, MulOpposite.algebraMap_apply]
state_before: case a.e_s.e_a.h
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
⊢ (∃ y, (algebraMap R A) y = MulOpposite.unop x) ↔ ∃ y, MulOpposite.op ((algebraMap R A) y) = x
state_after: case a.e_s.e_a.h.a.h.e'_2.h.a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
x✝ : R
⊢ (algebraMap R A) x✝ = MulOpposite.unop x ↔ MulOpposite.op ((algebraMap R A) x✝) = x
tactic: congr!
state_before: case a.e_s.e_a.h.a.h.e'_2.h.a
ι : Sort u_1
R : Type u_2
A : Type u_3
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
x : Aᵐᵒᵖ
x✝ : R
⊢ (algebraMap R A) x✝ = MulOpposite.unop x ↔ MulOpposite.op ((algebraMap R A) x✝) = x
state_after: no goals
tactic: rw [← MulOpposite.op_injective.eq_iff (b := x.unop), MulOpposite.op_unop]