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

Proof steps:
state_before: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
_RB : Algebra R B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
g₁ g₂ : A →ₐ[R] B
h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂
⊢ g₁ = g₂
state_after: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
_RB : Algebra R B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ (g₁ g₂ : A →ₐ[R] B), (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂ → g₁ = g₂
tactic: revert g₁ g₂
state_before: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
_RB : Algebra R B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ (g₁ g₂ : A →ₐ[R] B), (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂ → g₁ = g₂
state_after: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
_RB : Algebra R B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ Function.Injective (Ideal.Quotient.mkₐ R I).comp
tactic: change Function.Injective (Ideal.Quotient.mkₐ R I).comp
state_before: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
_RB : Algebra R B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ Function.Injective (Ideal.Quotient.mkₐ R I).comp
state_after: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ [_RB : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp
tactic: revert _RB
state_before: R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ [_RB : Algebra R B], Function.Injective (Ideal.Quotient.mkₐ R I).comp
state_after: case h₁
R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ ⦃S : Type u⦄ [inst : CommRing S] (I : Ideal S),
    I ^ 2 = ⊥ → ∀ [_RB : Algebra R S], Function.Injective (Ideal.Quotient.mkₐ R I).comp

case h₂
R : Type u
inst✝⁶ : CommSemiring R
A : Type u
inst✝⁵ : Semiring A
inst✝⁴ : Algebra R A
B✝ : Type u
inst✝³ : CommRing B✝
inst✝² : Algebra R B✝
I✝ : Ideal B✝
B : Type u
inst✝¹ : CommRing B
inst✝ : FormallyUnramified R A
I : Ideal B
hI : IsNilpotent I
⊢ ∀ ⦃S : Type u⦄ [inst : CommRing S] (I J : Ideal S),
    I ≤ J →
      (∀ [_RB : Algebra R S], Function.Injective (Ideal.Quotient.mkₐ R I).comp) →
        (∀ [_RB : Algebra R (S ⧸ I)],
            Function.Injective (Ideal.Quotient.mkₐ R (Ideal.map (Ideal.Quotient.mk I) J)).comp) →
          ∀ [_RB : Algebra R S], Function.Injective (Ideal.Quotient.mkₐ R J).comp
tactic: apply Ideal.IsNilpotent.induction_on (R := B) I hI