-- Mathlib/RingTheory/Ideal/GoingDown.lean
@[stacks 00HX]
lemma trans (T : Type*) [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
    [Algebra.HasGoingDown R S] [Algebra.HasGoingDown S T] :
    Algebra.HasGoingDown R T

Proof state:
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap ((algebraMap S T).comp (algebraMap R S)))

Tactic:
simp only [PrimeSpectrum.comap_comp, ContinuousMap.coe_comp]

Result:
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap (⇑(PrimeSpectrum.comap (algebraMap R S)) ∘ ⇑(PrimeSpectrum.comap (algebraMap S T)))

Proof state:
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap (⇑(PrimeSpectrum.comap (algebraMap R S)) ∘ ⇑(PrimeSpectrum.comap (algebraMap S T)))

Tactic:
apply GeneralizingMap.comp

Result:
case hf
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap (algebraMap S T))

case hg
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap (algebraMap R S))

Proof state:
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ HasGoingDown R T

Tactic:
rw [iff_generalizingMap_primeSpectrumComap, IsScalarTower.algebraMap_eq R S T]

Result:
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap ((algebraMap S T).comp (algebraMap R S)))

Proof state:
case hf
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap (algebraMap S T))

Tactic:
rwa [← iff_generalizingMap_primeSpectrumComap]

Result:
no goals

Proof state:
case hg
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingDown R S
inst✝ : HasGoingDown S T
⊢ GeneralizingMap ⇑(PrimeSpectrum.comap (algebraMap R S))

Tactic:
rwa [← iff_generalizingMap_primeSpectrumComap]

Result:
no goals