Theorem: Algebra.FormallyUnramified.of_equiv
File: Mathlib/RingTheory/Unramified/Basic.lean

Proof steps:
state_before: R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallyUnramified R A
e : A ≃ₐ[R] B
⊢ FormallyUnramified R B
state_after: case comp_injective
R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallyUnramified R A
e : A ≃ₐ[R] B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
tactic: constructor
state_before: case comp_injective
R : Type u
inst✝⁵ : CommSemiring R
A B : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : Semiring B
inst✝¹ : Algebra R B
inst✝ : FormallyUnramified R A
e : A ≃ₐ[R] B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),
    I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp
state_after: case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁ = f₂
tactic: intro C _ _ I hI f₁ f₂ e'
state_before: case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁ = f₂
state_after: case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ (f₁.comp ↑e).comp ↑e.symm = (f₂.comp ↑e).comp ↑e.symm
tactic: rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
state_before: case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ (f₁.comp ↑e).comp ↑e.symm = (f₂.comp ↑e).comp ↑e.symm
state_after: case comp_injective.e_φ₁
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁.comp ↑e = f₂.comp ↑e
tactic: congr 1
state_before: case comp_injective.e_φ₁
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ f₁.comp ↑e = f₂.comp ↑e
state_after: case comp_injective.e_φ₁
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ (Ideal.Quotient.mkₐ R I).comp (f₁.comp ↑e) = (Ideal.Quotient.mkₐ R I).comp (f₂.comp ↑e)
tactic: refine FormallyUnramified.comp_injective I hI ?_
state_before: case comp_injective.e_φ₁
R : Type u
inst✝⁷ : CommSemiring R
A B : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
e : A ≃ₐ[R] B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e' : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂
⊢ (Ideal.Quotient.mkₐ R I).comp (f₁.comp ↑e) = (Ideal.Quotient.mkₐ R I).comp (f₂.comp ↑e)
state_after: no goals
tactic: rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]