-- Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
/-- The ideal of `A ⊗[R] B` generated by `I` is the image of `A ⊗[R] I` -/
lemma Ideal.map_includeRight_eq (I : Ideal B) :
    (I.map (Algebra.TensorProduct.includeRight : B →ₐ[R] A ⊗[R] B)).restrictScalars R
      = LinearMap.range (LinearMap.lTensor A (Submodule.subtype (I.restrictScalars R)))

Proof state:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x : A ⊗[R] ↥(Submodule.restrictScalars R I)
⊢ (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =
    0 • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x

Tactic:
simp only [map_zero, smul_eq_mul, zero_mul]

Result:
no goals

Proof state:
case a.refine_3
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x : A ⊗[R] B
hx : x ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
⊢ ∀ (x y : A ⊗[R] B),
    x ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I) →
      y ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I) →
        (∃ y, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y = x) →
          (∃ y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = y) →
            ∃ y_1, (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 = x + y

Tactic:
rintro x y - - ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩

Result:
case a.refine_3.intro.refl.intro.refl
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x y : A ⊗[R] ↥(Submodule.restrictScalars R I)
⊢ ∃ y_1,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =
      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +
        (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y

Proof state:
case a.refine_4.intro.refl.tmul.add
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx✝ : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
a : A
b : B
x y : A ⊗[R] ↥(Submodule.restrictScalars R I)
hx :
  ∃ y,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x
hy :
  ∃ y_1,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y
⊢ ∃ y_1,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)

Tactic:
obtain ⟨x', hx'⟩ := hx

Result:
case a.refine_4.intro.refl.tmul.add.intro
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
a : A
b : B
x y : A ⊗[R] ↥(Submodule.restrictScalars R I)
hy :
  ∃ y_1,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y
x' : A ⊗[R] ↥(Submodule.restrictScalars R I)
hx' :
  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x' =
    a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x
⊢ ∃ y_1,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y_1 =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) (x + y)

Proof state:
case a.refine_4.intro.refl.zero
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x : A ⊗[R] ↥(Submodule.restrictScalars R I)
⊢ ∃ y,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =
      0 • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x

Tactic:
use 0

Result:
case h
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x : A ⊗[R] ↥(Submodule.restrictScalars R I)
⊢ (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) 0 =
    0 • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x

Proof state:
case a.intro.tmul
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
a : A
b : ↥(Submodule.restrictScalars R I)
⊢ a ⊗ₜ[R] ↑b = a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] ↑b

Tactic:
simp only [Submodule.coe_restrictScalars, Algebra.TensorProduct.tmul_mul_tmul,
  mul_one, one_mul]

Result:
no goals

Proof state:
case a.refine_4.intro.refl.tmul
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x✝ : A ⊗[R] B
hx : x✝ ∈ Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)
x : A ⊗[R] ↥(Submodule.restrictScalars R I)
a : A
b : B
⊢ ∃ y,
    (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y =
      a ⊗ₜ[R] b • (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x

Tactic:
induction x with
| zero =>
  use 0
  simp only [map_zero, smul_eq_mul, mul_zero]
| tmul x y =>
  use (a * x) ⊗ₜ[R] (b •y)
  simp only [LinearMap.lTensor_tmul, Submodule.coe_subtype, smul_eq_mul, tmul_mul_tmul]
  rfl
| add x y hx hy =>
  obtain ⟨x', hx'⟩ := hx
  obtain ⟨y', hy'⟩ := hy
  use x' + y'
  simp only [map_add, hx', smul_add, hy']

Result:
no goals

Proof state:
case a.intro.add
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x y : A ⊗[R] ↥(Submodule.restrictScalars R I)
hx :
  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x ∈
    (Submodule.restrictScalars R (map includeRight I)).carrier
hy :
  (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y ∈
    (Submodule.restrictScalars R (map includeRight I)).carrier
⊢ (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) x +
      (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype) y ∈
    (Submodule.restrictScalars R (map includeRight I)).carrier

Tactic:
apply Submodule.add_mem _ hx hy

Result:
no goals

Proof state:
case a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
a : A
b : ↥(Submodule.restrictScalars R I)
this : a ⊗ₜ[R] ↑b = a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] ↑b
⊢ ↑b ∈ I

Tactic:
exact Submodule.coe_mem b

Result:
no goals

Proof state:
case a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
⊢ (Submodule.restrictScalars R (map includeRight I)).carrier ≤
    (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier

Tactic:
intro x

Result:
case a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
x : A ⊗[R] B
⊢ x ∈ (Submodule.restrictScalars R (map includeRight I)).carrier →
    x ∈ (LinearMap.range (LinearMap.lTensor A (Submodule.restrictScalars R I).subtype)).carrier