Module: Mathlib.RingTheory.Unramified.Finite

Goal state before:
pp: case zero
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) 0 =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * 0)

pp: case tmul
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ ⊗ₜ[R] y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝ ⊗ₜ[R] y✝)

pp: case add
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S ⊗[R] S
a✝¹ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) x✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝)
a✝ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) y✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * y✝)
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ + y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * (x✝ + y✝))

Tactic:
· simp

Goal state after:
case tmul
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ ⊗ₜ[R] y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝ ⊗ₜ[R] y✝)

case add
R : Type u_1
S : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
M : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : Module S M
inst✝² : IsScalarTower R S M
inst✝¹ : FormallyUnramified R S
inst✝ : EssFiniteType R S
r : S
m : M
x✝ y✝ : S ⊗[R] S
a✝¹ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) x✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * x✝)
a✝ :
  (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) y✝ =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * y✝)
⊢ (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip (r • m))) (x✝ + y✝) =
    (TensorProduct.AlgebraTensorModule.map LinearMap.id ((lsmul R R M).toLinearMap.flip m)) (1 ⊗ₜ[R] r * (x✝ + y✝))