Theorem: PiTensorProduct.algebraMap_apply
File: Mathlib/RingTheory/PiTensorProduct.lean

tactic: change r • tprod R 1 = _
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ (algebraMap R' (⨂[R] (i : ι), A i)) r = (tprod R) (Pi.mulSingle i ((algebraMap R' (A i)) r))
state_after: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ r • (tprod R) 1 = (tprod R) (Pi.mulSingle i ((algebraMap R' (A i)) r))

tactic: have : Pi.mulSingle i (algebraMap R' (A i) r) = update (fun i ↦ 1) i (r • 1) := by
  rw [Algebra.algebraMap_eq_smul_one]; rfl
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ r • (tprod R) 1 = (tprod R) (Pi.mulSingle i ((algebraMap R' (A i)) r))
state_after: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
this : Pi.mulSingle i ((algebraMap R' (A i)) r) = update (fun i => 1) i (r • 1)
⊢ r • (tprod R) 1 = (tprod R) (Pi.mulSingle i ((algebraMap R' (A i)) r))

tactic: rw [this, ← smul_one_smul R r (1 : A i), MultilinearMap.map_smul, update_eq_self, smul_one_smul]
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
this : Pi.mulSingle i ((algebraMap R' (A i)) r) = update (fun i => 1) i (r • 1)
⊢ r • (tprod R) 1 = (tprod R) (Pi.mulSingle i ((algebraMap R' (A i)) r))
state_after: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
this : Pi.mulSingle i ((algebraMap R' (A i)) r) = update (fun i => 1) i (r • 1)
⊢ r • (tprod R) 1 = r • ⨂ₜ[R] (i : ι), 1

tactic: congr
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
this : Pi.mulSingle i ((algebraMap R' (A i)) r) = update (fun i => 1) i (r • 1)
⊢ r • (tprod R) 1 = r • ⨂ₜ[R] (i : ι), 1
state_after: no goals

tactic: rw [Algebra.algebraMap_eq_smul_one]
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ Pi.mulSingle i ((algebraMap R' (A i)) r) = update (fun i => 1) i (r • 1)
state_after: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ Pi.mulSingle i (r • 1) = update (fun i => 1) i (r • 1)

tactic: rfl
state_before: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring R
inst✝⁵ : (i : ι) → Semiring (A i)
inst✝⁴ : Algebra R' R
inst✝³ : (i : ι) → Algebra R (A i)
inst✝² : (i : ι) → Algebra R' (A i)
inst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
i : ι
inst✝ : DecidableEq ι
⊢ Pi.mulSingle i (r • 1) = update (fun i => 1) i (r • 1)
state_after: no goals