/-- The ideal of `A ⊗[R] B` generated by `I` is the image of `I ⊗[R] B` -/
lemma Ideal.map_includeLeft_eq (I : Ideal A) :
    (I.map (Algebra.TensorProduct.includeLeft : A →ₐ[R] A ⊗[R] B)).restrictScalars R
      = LinearMap.range (LinearMap.rTensor B (Submodule.subtype (I.restrictScalars R))) := by
  rw [← Submodule.carrier_inj]
  apply le_antisymm
  · intro x
    simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
      Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem, LinearMap.mem_range]
    intro hx
    rw [Ideal.map, ← submodule_span_eq] at hx
    refine' Submodule.span_induction hx _ _ _ _
    · intro x
      simp only [includeLeft_apply, Set.mem_image, SetLike.mem_coe]
      rintro ⟨y, hy, rfl⟩
      use ⟨y, hy⟩ ⊗ₜ[R] 1
      rfl
    · use 0
      simp only [map_zero]
    · rintro x y ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩
      use x + y
      simp only [map_add]
    · rintro a x ⟨x, hx, rfl⟩
      induction a using TensorProduct.induction_on with
      | zero =>
        use 0
        simp only [map_zero, smul_eq_mul, zero_mul]
      | tmul a b =>
        induction x using TensorProduct.induction_on 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.coeSubtype, 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']
      | add a b ha hb =>
        obtain ⟨x', ha'⟩ := ha
        obtain ⟨y', hb'⟩ := hb
        use x' + y'
        simp only [map_add, ha', add_smul, hb']

  · rintro x ⟨y, rfl⟩
    induction y using TensorProduct.induction_on with
    | zero =>
        rw [map_zero]
        apply zero_mem
    | tmul a b =>
        simp only [LinearMap.rTensor_tmul, Submodule.coeSubtype]
        suffices : (a : A) ⊗ₜ[R] b = ((1 : A) ⊗ₜ[R] b) * ((a : A) ⊗ₜ[R] (1 : B))
        simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
          Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem]
        rw [this]
        apply Ideal.mul_mem_left
        apply Ideal.mem_map_of_mem
        exact Submodule.coe_mem a
        simp only [Submodule.coe_restrictScalars, Algebra.TensorProduct.tmul_mul_tmul,
          mul_one, one_mul]
    | add x y hx hy =>
        rw [map_add]
        

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 A
x y : ↥(Submodule.restrictScalars R I) ⊗[R] B
hx :
  (LinearMap.rTensor B (Submodule.subtype (Submodule.restrictScalars R I))) x ∈
    (Submodule.restrictScalars R (map includeLeft I)).toAddSubmonoid.toAddSubsemigroup.carrier
hy :
  (LinearMap.rTensor B (Submodule.subtype (Submodule.restrictScalars R I))) y ∈
    (Submodule.restrictScalars R (map includeLeft I)).toAddSubmonoid.toAddSubsemigroup.carrier
⊢ (LinearMap.rTensor B (Submodule.subtype (Submodule.restrictScalars R I))) x +
      (LinearMap.rTensor B (Submodule.subtype (Submodule.restrictScalars R I))) y ∈
    (Submodule.restrictScalars R (map includeLeft I)).toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
apply Submodule.add_mem _ hx hy