mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
a : A
x✝ : S
y✝ : Ω[A⁄R]
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • x✝ ⊗ₜ[R] y✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) (x✝ ⊗ₜ[R] y✝)

pp: case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
a : A
x✝ y✝ : S ⊗[R] Ω[A⁄R]
a✝¹ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • x✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) x✝
a✝ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • y✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) y✝
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • (x✝ + y✝)) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) (x✝ + y✝)

Tactic:
·

Goal state after:
case tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
a : A
x✝ : S
y✝ : Ω[A⁄R]
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • x✝ ⊗ₜ[R] y✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) (x✝ ⊗ₜ[R] y✝)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
a : A
x✝ y✝ : S ⊗[R] Ω[A⁄R]
a✝¹ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • x✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) x✝
a✝ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • y✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) y✝
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • (x✝ + y✝)) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) (x✝ + y✝)

Tactic:
·

Goal state after:
case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
a : A
x✝ y✝ : S ⊗[R] Ω[A⁄R]
a✝¹ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • x✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) x✝
a✝ :
  (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • y✝) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) y✝
⊢ (LinearMap.liftBaseChange S (↑R (map R S A B))) ((IsScalarTower.toAlgHom R A B).toLinearMap a • (x✝ + y✝)) =
    (IsScalarTower.toAlgHom R A B).toLinearMap a • (LinearMap.liftBaseChange S (↑R (map R S A B))) (x✝ + y✝)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mpr
K : Type u_1
A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)) → FormallyEtale K A

Tactic:
·

Goal state after:
case mpr
K : Type u_1
A : Type u
inst✝³ : Field K
inst✝² : CommRing A
inst✝¹ : Algebra K A
inst✝ : EssFiniteType K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Algebra.IsSeparable K (Ai i)) → FormallyEtale K A
Module: Mathlib.RingTheory.Etale.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ ↑(PrimeSpectrum.basicOpen f) ⊆ (Module.support A Ω[A⁄R])ᶜ ∧
      ↑(PrimeSpectrum.basicOpen f) ⊆ (Module.support A (H1Cotangent R A))ᶜ ↔
    FormallyEtale R (Localization.Away f)

Tactic:
Set.subset_compl_comm,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ Module.support A Ω[A⁄R] ⊆ (↑(PrimeSpectrum.basicOpen f))ᶜ ∧
      ↑(PrimeSpectrum.basicOpen f) ⊆ (Module.support A (H1Cotangent R A))ᶜ ↔
    FormallyEtale R (Localization.Away f)
Module: Mathlib.RingTheory.Etale.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ Module.support A Ω[A⁄R] ⊆ PrimeSpectrum.zeroLocus {f} ∧
      (PrimeSpectrum.zeroLocus {f})ᶜ ⊆ (Module.support A (H1Cotangent R A))ᶜ ↔
    FormallyEtale R (Localization.Away f)

Tactic:
Set.subset_compl_comm,

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ Module.support A Ω[A⁄R] ⊆ PrimeSpectrum.zeroLocus {f} ∧
      Module.support A (H1Cotangent R A) ⊆ (PrimeSpectrum.zeroLocus {f})ᶜᶜ ↔
    FormallyEtale R (Localization.Away f)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mp
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)

pp: case mpr
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)) → Etale K A

Tactic:
·

Goal state after:
case mp
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mp
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)

pp: case mpr
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)) → Etale K A

Tactic:
· intro H
    obtain ⟨I, _, Ai, _, _, e, _⟩ := (FormallyEtale.iff_exists_algEquiv_prod K A).mp inferInstance
    have := FormallyUnramified.finite_of_free K A
    exact ⟨_, ‹_›, _, _, _, e, fun i ↦ ⟨.of_surjective ((LinearMap.proj i).comp e.toLinearMap)
      ((Function.surjective_eval i).comp e.surjective), inferInstance⟩⟩

Goal state after:
case mpr
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)) → Etale K A
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case mpr
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)) → Etale K A

Tactic:
·

Goal state after:
case mpr
K : Type u_1
A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)) → Etale K A
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo 0) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo 0)

pp: case tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝))

pp: case add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A ⊗[R] A
a✝¹ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo x✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo x✝)
a✝ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo y✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo y✝)
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝))

Tactic:
·

Goal state after:
case zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo 0) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo 0)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝))

pp: case add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A ⊗[R] A
a✝¹ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo x✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo x✝)
a✝ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo y✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo y✝)
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝))

Tactic:
·

Goal state after:
case tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ ⊗ₜ[R] y✝))
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A ⊗[R] A
a✝¹ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo x✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo x✝)
a✝ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo y✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo y✝)
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝))

Tactic:
·

Goal state after:
case add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : Algebra S B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : Algebra.IsPushout R S A B
x : S
x✝ y✝ : A ⊗[R] A
a✝¹ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo x✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo x✝)
a✝ :
  (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo y✝) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo y✝)
⊢ (↑R (↑S (derivationTensorProduct R S A B).liftKaehlerDifferential ∘ₗ LinearMap.liftBaseChange S (↑R (map R S A B))))
      (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝)) =
    (↑R LinearMap.id) (x ⊗ₜ[R] (D R A).tensorProductTo (x✝ + y✝))
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) 0) = x • (D S B) 0

pp: case tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
m✝ : A
⊢ x •
      (LinearMap.liftBaseChange S (↑R (map R S A B)))
        ((derivationTensorProduct R S A B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)) =
    x • (D S B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)

pp: case tmul.smul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
s✝ : S
n✝ : B
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n✝) = x • (D S B) n✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (s✝ • n✝)) =
    x • (D S B) (s✝ • n✝)

pp: case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x n₁✝ n₂✝ : B
a✝¹ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₁✝) = x • (D S B) n₁✝
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₂✝) = x • (D S B) n₂✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (n₁✝ + n₂✝)) =
    x • (D S B) (n₁✝ + n₂✝)

Tactic:
·

Goal state after:
case tmul.zero
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) 0) = x • (D S B) 0
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
m✝ : A
⊢ x •
      (LinearMap.liftBaseChange S (↑R (map R S A B)))
        ((derivationTensorProduct R S A B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)) =
    x • (D S B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)

pp: case tmul.smul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
s✝ : S
n✝ : B
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n✝) = x • (D S B) n✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (s✝ • n✝)) =
    x • (D S B) (s✝ • n✝)

pp: case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x n₁✝ n₂✝ : B
a✝¹ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₁✝) = x • (D S B) n₁✝
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₂✝) = x • (D S B) n₂✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (n₁✝ + n₂✝)) =
    x • (D S B) (n₁✝ + n₂✝)

Tactic:
·

Goal state after:
case tmul.tmul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
m✝ : A
⊢ x •
      (LinearMap.liftBaseChange S (↑R (map R S A B)))
        ((derivationTensorProduct R S A B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)) =
    x • (D S B) ((IsScalarTower.toAlgHom R A B).toLinearMap m✝)
Module: Mathlib.RingTheory.Kaehler.TensorProduct

Goal state before:
pp: case tmul.smul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
s✝ : S
n✝ : B
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n✝) = x • (D S B) n✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (s✝ • n✝)) =
    x • (D S B) (s✝ • n✝)

pp: case tmul.add
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x n₁✝ n₂✝ : B
a✝¹ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₁✝) = x • (D S B) n₁✝
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n₂✝) = x • (D S B) n₂✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (n₁✝ + n₂✝)) =
    x • (D S B) (n₁✝ + n₂✝)

Tactic:
·

Goal state after:
case tmul.smul
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
inst✝⁷ : CommRing A
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : Algebra S B
inst✝¹ : IsScalarTower R A B
inst✝ : IsScalarTower R S B
h : Algebra.IsPushout R S A B
x : B
s✝ : S
n✝ : B
a✝ : x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) n✝) = x • (D S B) n✝
⊢ x • (LinearMap.liftBaseChange S (↑R (map R S A B))) ((derivationTensorProduct R S A B) (s✝ • n✝)) =
    x • (D S B) (s✝ • n✝)