Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (equivQuotMaximalIdeal p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ))
        ((equivQuotientMapMaximalIdeal S p Rₚ Sₚ) ((Ideal.Quotient.mk pS) x))) =
    (equivQuotMaximalIdeal p Rₚ).symm ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

pp: case he
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdeal p Rₚ) =
    (↑(equivQuotientMapMaximalIdeal S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (equivQuotMaximalIdeal p Rₚ).symm
      ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ))
        ((equivQuotientMapMaximalIdeal S p Rₚ Sₚ) ((Ideal.Quotient.mk pS) x))) =
    (equivQuotMaximalIdeal p Rₚ).symm ((Algebra.trace (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)) ((algebraMap S (Sₚ ⧸ pSₚ)) x))

case he
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
p : Ideal R
inst✝¹¹ : p.IsMaximal
Rₚ : Type u_3
Sₚ : Type u_4
inst✝¹⁰ : CommRing Rₚ
inst✝⁹ : CommRing Sₚ
inst✝⁸ : Algebra R Rₚ
inst✝⁷ : IsLocalization.AtPrime Rₚ p
inst✝⁶ : IsLocalRing Rₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : Algebra R Sₚ
inst✝³ : Algebra Rₚ Sₚ
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝¹ : IsScalarTower R S Sₚ
inst✝ : IsScalarTower R Rₚ Sₚ
x : S
this : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)
⊢ (algebraMap (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ)).comp ↑(equivQuotMaximalIdeal p Rₚ) =
    (↑(equivQuotientMapMaximalIdeal S p Rₚ Sₚ)).comp (algebraMap (R ⧸ p) (S ⧸ pS))