Theorem: DividedPowerAlgebra.condτ_rel (DividedPowers/DPAlgebra/Dpow.lean)

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hRS : Condτ A hI hJ
⊢ Condτ A hI' hJ'
Tactic:
obtain ⟨hK, hK_pd⟩ := hRS
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
⊢ Condτ A hI' hJ'

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
⊢ Condτ A hI' hJ'
Tactic:
simp only [Condτ]
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')
Tactic:
let fg := Algebra.TensorProduct.map f g
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')
Tactic:
have s_fg : Function.Surjective fg := TensorProduct.map_surjective hf hg
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')
Tactic:
have hK_map : K A I' J' = (K A I J).map fg := by
  simp only [K, fg, hI'I, hJ'J]
  rw [Ideal.map_sup]
  apply congr_arg₂
  all_goals
    simp only [Ideal.map_toRingHom, Ideal.map_map]
    apply congr_arg₂ _ _ rfl
    ext x
    simp only [i1, i2, RingHom.comp_apply, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
      Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.includeRight_apply, Algebra.TensorProduct.map_tmul, map_one]
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')
Tactic:
have hK'_pd : isSubDPIdeal hK (RingHom.ker fg ⊓ K A I J) := by
  rw [roby]
  apply isSubDPIdeal_sup
  exact isSubDPIdeal_map hI hK hK_pd.1 _ (isSubDPIdeal_ker hI hI' hf')
  exact isSubDPIdeal_map hJ hK hK_pd.2 _ (isSubDPIdeal_ker hJ hJ' hg')
State after:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')

State before:
case intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∃ hK, hI'.isDPMorphism hK ↑(i1 A R' S') ∧ hJ'.isDPMorphism hK ↑(i2 A R' S')
Tactic:
use DividedPowers.Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd
State after:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hI'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i1 A R' S') ∧
    hJ'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i2 A R' S')

State before:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hI'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i1 A R' S') ∧
    hJ'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i2 A R' S')
Tactic:
constructor
State after:
case h.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hI'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i1 A R' S')

case h.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hJ'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i2 A R' S')

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ K A I' J' = Ideal.map fg (K A I J)
Tactic:
simp only [K, fg, hI'I, hJ'J]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) ⊔ Ideal.map (i2 A R' S') (Ideal.map g J) =
    Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I ⊔ Ideal.map (i2 A R S) J)

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) ⊔ Ideal.map (i2 A R' S') (Ideal.map g J) =
    Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I ⊔ Ideal.map (i2 A R S) J)
Tactic:
rw [Ideal.map_sup]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) ⊔ Ideal.map (i2 A R' S') (Ideal.map g J) =
    Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I) ⊔
      Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i2 A R S) J)

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) ⊔ Ideal.map (i2 A R' S') (Ideal.map g J) =
    Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I) ⊔
      Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i2 A R S) J)
Tactic:
apply congr_arg₂
State after:
case hx
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) = Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I)

case hy
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i2 A R' S') (Ideal.map g J) = Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i2 A R S) J)

State before:
case hx
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i1 A R' S') (Ideal.map f I) = Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i1 A R S) I)

case hy
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i2 A R' S') (Ideal.map g J) = Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i2 A R S) J)
Tactic:
all_goals
  simp only [Ideal.map_toRingHom, Ideal.map_map]
  apply congr_arg₂ _ _ rfl
  ext x
  simp only [i1, i2, RingHom.comp_apply, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
    Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.includeRight_apply, Algebra.TensorProduct.map_tmul, map_one]
State after:
no goals

State before:
case hy
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map (i2 A R' S') (Ideal.map g J) = Ideal.map (Algebra.TensorProduct.map f g) (Ideal.map (i2 A R S) J)
Tactic:
simp only [Ideal.map_toRingHom, Ideal.map_map]
State after:
case hy
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map ((i2 A R' S').comp g.toRingHom) J = Ideal.map ((Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom) J

State before:
case hy
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ Ideal.map ((i2 A R' S').comp g.toRingHom) J = Ideal.map ((Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom) J
Tactic:
apply congr_arg₂ _ _ rfl
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ (i2 A R' S').comp g.toRingHom = (Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
⊢ (i2 A R' S').comp g.toRingHom = (Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom
Tactic:
ext x
State after:
case a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
x : S
⊢ ((i2 A R' S').comp g.toRingHom) x = ((Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom) x

State before:
case a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
x : S
⊢ ((i2 A R' S').comp g.toRingHom) x = ((Algebra.TensorProduct.map f g).comp (i2 A R S).toRingHom) x
Tactic:
simp only [i1, i2, RingHom.comp_apply, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
  Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.includeRight_apply, Algebra.TensorProduct.map_tmul, map_one]
State after:
no goals

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
Tactic:
rw [roby]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal
    (Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal
    (Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))
Tactic:
apply isSubDPIdeal_sup
State after:
case hJ
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I))

case hK
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))

State before:
case hJ
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I))

case hK
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))
Tactic:
exact isSubDPIdeal_map hI hK hK_pd.1 _ (isSubDPIdeal_ker hI hI' hf')
State after:
case hK
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))

State before:
case hK
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
⊢ hK.isSubDPIdeal (Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J))
Tactic:
exact isSubDPIdeal_map hJ hK hK_pd.2 _ (isSubDPIdeal_ker hJ hJ' hg')
State after:
no goals

State before:
case h.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hI'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i1 A R' S')
Tactic:
constructor
State after:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i1 A R' S')) I' ≤ Ideal.map fg (K A I J)

case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∀ (n : ℕ),
    ∀ a ∈ I',
      (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a) = ↑(i1 A R' S') (hI'.dpow n a)

State before:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i1 A R' S')) I' ≤ Ideal.map fg (K A I J)
Tactic:
rw [← hK_map]
State after:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i1 A R' S')) I' ≤ K A I' J'

State before:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i1 A R' S')) I' ≤ K A I' J'
Tactic:
rw [Ideal.map_le_iff_le_comap]
State after:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ I' ≤ comap (↑(i1 A R' S')) (K A I' J')

State before:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ I' ≤ comap (↑(i1 A R' S')) (K A I' J')
Tactic:
intro a' ha'
State after:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ a' ∈ comap (↑(i1 A R' S')) (K A I' J')

State before:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ a' ∈ comap (↑(i1 A R' S')) (K A I' J')
Tactic:
rw [Ideal.mem_comap]
State after:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ ↑(i1 A R' S') a' ∈ K A I' J'

State before:
case h.left.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ ↑(i1 A R' S') a' ∈ K A I' J'
Tactic:
apply Ideal.mem_sup_left
State after:
case h.left.left.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ ↑(i1 A R' S') a' ∈ Ideal.map (i1 A R' S') I'

State before:
case h.left.left.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ ↑(i1 A R' S') a' ∈ Ideal.map (i1 A R' S') I'
Tactic:
apply Ideal.mem_map_of_mem
State after:
case h.left.left.a.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ a' ∈ I'

State before:
case h.left.left.a.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : R'
ha' : a' ∈ I'
⊢ a' ∈ I'
Tactic:
exact ha'
State after:
no goals

State before:
case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∀ (n : ℕ),
    ∀ a ∈ I',
      (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a) = ↑(i1 A R' S') (hI'.dpow n a)
Tactic:
intro n a' ha'
State after:
case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : R'
ha' : a' ∈ I'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a') = ↑(i1 A R' S') (hI'.dpow n a')

State before:
case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : R'
ha' : a' ∈ I'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a') = ↑(i1 A R' S') (hI'.dpow n a')
Tactic:
simp only [hI'I, Ideal.mem_map_iff_of_surjective f hf] at ha'
State after:
case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : R'
ha' : ∃ x ∈ I, f x = a'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a') = ↑(i1 A R' S') (hI'.dpow n a')

State before:
case h.left.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : R'
ha' : ∃ x ∈ I, f x = a'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') a') = ↑(i1 A R' S') (hI'.dpow n a')
Tactic:
obtain ⟨a, ha, rfl⟩ := ha'
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') (f a)) = ↑(i1 A R' S') (hI'.dpow n (f a))

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i1 A R' S') (f a)) = ↑(i1 A R' S') (hI'.dpow n (f a))
Tactic:
simp only [i1, AlgHom.coe_toRingHom, Algebra.TensorProduct.includeLeft_apply]
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (f a ⊗ₜ[A] 1) = hI'.dpow n (f a) ⊗ₜ[A] 1

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (f a ⊗ₜ[A] 1) = hI'.dpow n (f a) ⊗ₜ[A] 1
Tactic:
rw [← map_one g, ← Algebra.TensorProduct.map_tmul]
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    hI'.dpow n (f a) ⊗ₜ[A] g 1

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    hI'.dpow n (f a) ⊗ₜ[A] g 1
Tactic:
rw [← AlgHom.coe_toRingHom f, hf'.2 n a ha, RingHom.coe_coe]
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    f (hI.dpow n a) ⊗ₜ[A] g 1

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    f (hI.dpow n a) ⊗ₜ[A] g 1
Tactic:
rw [← Algebra.TensorProduct.map_tmul]
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    (Algebra.TensorProduct.map f g) (hI.dpow n a ⊗ₜ[A] 1)

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n ((Algebra.TensorProduct.map f g) (a ⊗ₜ[A] 1)) =
    (Algebra.TensorProduct.map f g) (hI.dpow n a ⊗ₜ[A] 1)
Tactic:
erw [Quotient.OfSurjective.dpow_apply hK s_fg hK'_pd]
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ fg.toRingHom (hK.dpow n (a ⊗ₜ[A] 1)) = (Algebra.TensorProduct.map f g) (hI.dpow n a ⊗ₜ[A] 1)

case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ fg.toRingHom (hK.dpow n (a ⊗ₜ[A] 1)) = (Algebra.TensorProduct.map f g) (hI.dpow n a ⊗ₜ[A] 1)

case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J
Tactic:
apply congr_arg
State after:
case h.left.right.intro.intro.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ hK.dpow n (a ⊗ₜ[A] 1) = hI.dpow n a ⊗ₜ[A] 1

case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J

State before:
case h.left.right.intro.intro.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ hK.dpow n (a ⊗ₜ[A] 1) = hI.dpow n a ⊗ₜ[A] 1

case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J
Tactic:
exact hK_pd.1.2 n a ha
State after:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J

State before:
case h.left.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ K A I J
Tactic:
apply Ideal.mem_sup_left
State after:
case h.left.right.intro.intro.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ Ideal.map (i1 A R S) I

State before:
case h.left.right.intro.intro.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : R
ha : a ∈ I
⊢ a ⊗ₜ[A] 1 ∈ Ideal.map (i1 A R S) I
Tactic:
apply Ideal.mem_map_of_mem _ ha
State after:
no goals

State before:
case h.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ hJ'.isDPMorphism (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd) ↑(i2 A R' S')
Tactic:
constructor
State after:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i2 A R' S')) J' ≤ Ideal.map fg (K A I J)

case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∀ (n : ℕ),
    ∀ a ∈ J',
      (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a) = ↑(i2 A R' S') (hJ'.dpow n a)

State before:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i2 A R' S')) J' ≤ Ideal.map fg (K A I J)
Tactic:
rw [← hK_map]
State after:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i2 A R' S')) J' ≤ K A I' J'

State before:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ Ideal.map (↑(i2 A R' S')) J' ≤ K A I' J'
Tactic:
rw [Ideal.map_le_iff_le_comap]
State after:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ J' ≤ comap (↑(i2 A R' S')) (K A I' J')

State before:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ J' ≤ comap (↑(i2 A R' S')) (K A I' J')
Tactic:
intro a' ha'
State after:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ a' ∈ comap (↑(i2 A R' S')) (K A I' J')

State before:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ a' ∈ comap (↑(i2 A R' S')) (K A I' J')
Tactic:
rw [Ideal.mem_comap]
State after:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ ↑(i2 A R' S') a' ∈ K A I' J'

State before:
case h.right.left
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ ↑(i2 A R' S') a' ∈ K A I' J'
Tactic:
apply Ideal.mem_sup_right
State after:
case h.right.left.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ ↑(i2 A R' S') a' ∈ Ideal.map (i2 A R' S') J'

State before:
case h.right.left.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ ↑(i2 A R' S') a' ∈ Ideal.map (i2 A R' S') J'
Tactic:
apply Ideal.mem_map_of_mem
State after:
case h.right.left.a.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ a' ∈ J'

State before:
case h.right.left.a.h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
a' : S'
ha' : a' ∈ J'
⊢ a' ∈ J'
Tactic:
exact ha'
State after:
no goals

State before:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
⊢ ∀ (n : ℕ),
    ∀ a ∈ J',
      (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a) = ↑(i2 A R' S') (hJ'.dpow n a)
Tactic:
intro n a' ha'
State after:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : S'
ha' : a' ∈ J'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a') = ↑(i2 A R' S') (hJ'.dpow n a')

State before:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : S'
ha' : a' ∈ J'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a') = ↑(i2 A R' S') (hJ'.dpow n a')
Tactic:
simp only [hJ'J, Ideal.mem_map_iff_of_surjective g hg] at ha'
State after:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : S'
ha' : ∃ x ∈ J, g x = a'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a') = ↑(i2 A R' S') (hJ'.dpow n a')

State before:
case h.right.right
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a' : S'
ha' : ∃ x ∈ J, g x = a'
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') a') = ↑(i2 A R' S') (hJ'.dpow n a')
Tactic:
obtain ⟨a, ha, rfl⟩ := ha'
State after:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') (g a)) = ↑(i2 A R' S') (hJ'.dpow n (g a))

State before:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (↑(i2 A R' S') (g a)) = ↑(i2 A R' S') (hJ'.dpow n (g a))
Tactic:
simp only [i2, AlgHom.coe_toRingHom, Algebra.TensorProduct.includeRight_apply]
State after:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (1 ⊗ₜ[A] g a) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

State before:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (1 ⊗ₜ[A] g a) = 1 ⊗ₜ[A] hJ'.dpow n (g a)
Tactic:
suffices ∀ y : S, fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y by
  rw [← this]
  rw [Quotient.OfSurjective.dpow_apply hK s_fg]
  have that := hg'.2 n a ha
  simp only [AlgHom.coe_toRingHom] at that ; rw [that]
  rw [← this]
  apply congr_arg
  simp only [← Algebra.TensorProduct.includeRight_apply]
  exact hK_pd.2.2 n a ha
  apply Ideal.mem_sup_right
  apply Ideal.mem_map_of_mem _ ha
State after:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y

State before:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
⊢ ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
Tactic:
intro x
State after:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
x : S
⊢ fg.toRingHom (1 ⊗ₜ[A] x) = 1 ⊗ₜ[A] g x

State before:
case h.right.right.intro.intro
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
x : S
⊢ fg.toRingHom (1 ⊗ₜ[A] x) = 1 ⊗ₜ[A] g x
Tactic:
simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, Algebra.TensorProduct.map_tmul, map_one,
  fg]
State after:
no goals

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (1 ⊗ₜ[A] g a) = 1 ⊗ₜ[A] hJ'.dpow n (g a)
Tactic:
rw [← this]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (fg.toRingHom (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ (Quotient.OfSurjective.dividedPowers hK s_fg hK'_pd).dpow n (fg.toRingHom (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)
Tactic:
rw [Quotient.OfSurjective.dpow_apply hK s_fg]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
have that := hg'.2 n a ha
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (↑g a) = ↑g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (↑g a) = ↑g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
simp only [AlgHom.coe_toRingHom] at that
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] hJ'.dpow n (g a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
rw [that]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] g (hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = 1 ⊗ₜ[A] g (hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
rw [← this]
State after:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = fg.toRingHom (1 ⊗ₜ[A] hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ fg.toRingHom (hK.dpow n (1 ⊗ₜ[A] a)) = fg.toRingHom (1 ⊗ₜ[A] hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
apply congr_arg
State after:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ hK.dpow n (1 ⊗ₜ[A] a) = 1 ⊗ₜ[A] hJ.dpow n a

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ hK.dpow n (1 ⊗ₜ[A] a) = 1 ⊗ₜ[A] hJ.dpow n a

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
simp only [← Algebra.TensorProduct.includeRight_apply]
State after:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ hK.dpow n (Algebra.TensorProduct.includeRight a) = Algebra.TensorProduct.includeRight (hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
case h
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
that : hJ'.dpow n (g a) = g (hJ.dpow n a)
⊢ hK.dpow n (Algebra.TensorProduct.includeRight a) = Algebra.TensorProduct.includeRight (hJ.dpow n a)

case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
exact hK_pd.2.2 n a ha
State after:
case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J

State before:
case ha
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ K A I J
Tactic:
apply Ideal.mem_sup_right
State after:
case ha.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ Ideal.map (i2 A R S) J

State before:
case ha.a
R✝ : Type u
inst✝⁹ : CommRing R✝
A : Type u
inst✝⁸ : CommRing A
R S R' S' : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra A R
inst✝² : Algebra A S
inst✝¹ : Algebra A R'
inst✝ : Algebra A S'
f : R →ₐ[A] R'
hf : Function.Surjective ⇑f
I : Ideal R
hI : DividedPowers I
I' : Ideal R'
hI' : DividedPowers I'
hf' : hI.isDPMorphism hI' ↑f
hI'I : I' = Ideal.map f I
g : S →ₐ[A] S'
hg : Function.Surjective ⇑g
J : Ideal S
hJ : DividedPowers J
J' : Ideal S'
hJ' : DividedPowers J'
hg' : hJ.isDPMorphism hJ' ↑g
hJ'J : J' = Ideal.map g J
roby :
  RingHom.ker (Algebra.TensorProduct.map f g) ⊓ K A I J =
    Ideal.map Algebra.TensorProduct.includeLeft (RingHom.ker f ⊓ I) ⊔
      Ideal.map Algebra.TensorProduct.includeRight (RingHom.ker g ⊓ J)
hK : DividedPowers (K A I J)
hK_pd : hI.isDPMorphism hK ↑(i1 A R S) ∧ hJ.isDPMorphism hK ↑(i2 A R S)
fg : R ⊗[A] S →ₐ[A] R' ⊗[A] S' := Algebra.TensorProduct.map f g
s_fg : Function.Surjective ⇑fg
hK_map : K A I' J' = Ideal.map fg (K A I J)
hK'_pd : hK.isSubDPIdeal (RingHom.ker fg ⊓ K A I J)
n : ℕ
a : S
ha : a ∈ J
this : ∀ (y : S), fg.toRingHom (1 ⊗ₜ[A] y) = 1 ⊗ₜ[A] g y
⊢ 1 ⊗ₜ[A] a ∈ Ideal.map (i2 A R S) J
Tactic:
apply Ideal.mem_map_of_mem _ ha
State after:
no goals