Module: Mathlib.LinearAlgebra.TensorProduct.RightExactness

Goal state before:
pp: case a.refine_1
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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∀ x ∈ ⇑includeLeft '' ↑I, ∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = x

pp: case a.refine_2
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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = 0

pp: 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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∀ (x y : A ⊗[R] B),
    x ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
      y ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
        (∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = x) →
          (∃ y_1, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y_1 = y) →
            ∃ y_1, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y_1 = x + y

pp: case a.refine_4
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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∀ (a x : A ⊗[R] B),
    x ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
      (∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = x) →
        ∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = a • x

Tactic:
· intro x
      simp only [includeLeft_apply, Set.mem_image, SetLike.mem_coe]
      rintro ⟨y, hy, rfl⟩
      use ⟨y, hy⟩ ⊗ₜ[R] 1
      rfl

Goal state after:
case a.refine_2
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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = 0

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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∀ (x y : A ⊗[R] B),
    x ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
      y ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
        (∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = x) →
          (∃ y_1, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y_1 = y) →
            ∃ y_1, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y_1 = x + y

case a.refine_4
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 A
x : A ⊗[R] B
hx : x ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))
⊢ ∀ (a x : A ⊗[R] B),
    x ∈ Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I) →
      (∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = x) →
        ∃ y, (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype) y = a • x