Theorem: Algebra.TensorProduct.ext
File: Mathlib/RingTheory/TensorProduct/Basic.lean

Proof steps:
state_before: R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
⊢ f = g
state_after: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
⊢ f.toLinearMap = g.toLinearMap
tactic: apply AlgHom.toLinearMap_injective
state_before: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
⊢ f.toLinearMap = g.toLinearMap
state_after: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b
tactic: ext a b
state_before: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b
state_after: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this :
  (f.comp includeLeft) a * ((AlgHom.restrictScalars R f).comp includeRight) b =
    (g.comp includeLeft) a * ((AlgHom.restrictScalars R g).comp includeRight) b
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b
tactic: have := congr_arg₂ HMul.hMul (AlgHom.congr_fun ha a) (AlgHom.congr_fun hb b)
state_before: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this :
  (f.comp includeLeft) a * ((AlgHom.restrictScalars R f).comp includeRight) b =
    (g.comp includeLeft) a * ((AlgHom.restrictScalars R g).comp includeRight) b
⊢ ((AlgebraTensorModule.curry f.toLinearMap) a) b = ((AlgebraTensorModule.curry g.toLinearMap) a) b
state_after: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this : f (a ⊗ₜ[R] 1) * f (1 ⊗ₜ[R] b) = g (a ⊗ₜ[R] 1) * g (1 ⊗ₜ[R] b)
⊢ f (a ⊗ₜ[R] b) = g (a ⊗ₜ[R] b)
tactic: dsimp at *
state_before: case a.a.h.h
R : Type uR
S : Type uS
A : Type uA
B : Type uB
C : Type uC
D : Type uD
E : Type uE
F : Type uF
inst✝¹² : CommSemiring R
inst✝¹¹ : Semiring A
inst✝¹⁰ : Algebra R A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R B
inst✝⁷ : Semiring C
inst✝⁶ : Algebra R C
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S A
inst✝³ : Algebra R S
inst✝² : Algebra S C
inst✝¹ : IsScalarTower R S A
inst✝ : IsScalarTower R S C
f g : A ⊗[R] B →ₐ[S] C
ha : f.comp includeLeft = g.comp includeLeft
hb : (AlgHom.restrictScalars R f).comp includeRight = (AlgHom.restrictScalars R g).comp includeRight
a : A
b : B
this : f (a ⊗ₜ[R] 1) * f (1 ⊗ₜ[R] b) = g (a ⊗ₜ[R] 1) * g (1 ⊗ₜ[R] b)
⊢ f (a ⊗ₜ[R] b) = g (a ⊗ₜ[R] b)
state_after: no goals
tactic: rwa [← _root_.map_mul, ← _root_.map_mul, tmul_mul_tmul, _root_.one_mul, _root_.mul_one] at this