Theorem: RingHom.SurjectiveOnStalks.of_comp
File: Mathlib/RingTheory/SurjectiveOnStalks.lean

Proof steps:
state_before: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
⊢ g.SurjectiveOnStalks
state_after: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
tactic: intros I hI
state_before: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
state_after: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
this : Function.Surjective ⇑(Localization.localRingHom (Ideal.comap (g.comp f) I) I (g.comp f) ⋯)
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
tactic: have := hg I hI
state_before: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
this : Function.Surjective ⇑(Localization.localRingHom (Ideal.comap (g.comp f) I) I (g.comp f) ⋯)
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
state_after: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
this :
  Function.Surjective
    (⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯) ∘
      ⇑(Localization.localRingHom (Ideal.comap (g.comp f) I) (Ideal.comap g I) f ⋯))
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
tactic: rw [Localization.localRingHom_comp (I.comap (g.comp f)) (I.comap g) _ _ rfl _ rfl,
    RingHom.coe_comp] at this
state_before: R : Type u_1
inst✝⁶ : CommRing R
M : Submonoid R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
g : S →+* T
f : R →+* S
hg : (g.comp f).SurjectiveOnStalks
I : Ideal T
hI : I.IsPrime
this :
  Function.Surjective
    (⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯) ∘
      ⇑(Localization.localRingHom (Ideal.comap (g.comp f) I) (Ideal.comap g I) f ⋯))
⊢ Function.Surjective ⇑(Localization.localRingHom (Ideal.comap g I) I g ⋯)
state_after: no goals
tactic: exact this.of_comp