Metamath Proof Explorer

Theorem mulvfv

Description: Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012)

Ref Expression
Assertion mulvfv ⊢ A ∈ E ∧ B ∈ D ∧ C ∈ ℝ → A ⋅ v B ⁡ C = A ⁢ B ⁡ C

Proof

Step Hyp Ref Expression
1 mulvval ⊢ A ∈ E ∧ B ∈ D → A ⋅ v B = x ∈ ℝ ⟼ A ⁢ B ⁡ x
2 1 fveq1d ⊢ A ∈ E ∧ B ∈ D → A ⋅ v B ⁡ C = x ∈ ℝ ⟼ A ⁢ B ⁡ x ⁡ C
3 fveq2 ⊢ x = C → B ⁡ x = B ⁡ C
4 3 oveq2d ⊢ x = C → A ⁢ B ⁡ x = A ⁢ B ⁡ C
5 eqid ⊢ x ∈ ℝ ⟼ A ⁢ B ⁡ x = x ∈ ℝ ⟼ A ⁢ B ⁡ x
6 ovex ⊢ A ⁢ B ⁡ C ∈ V
7 4 5 6 fvmpt ⊢ C ∈ ℝ → x ∈ ℝ ⟼ A ⁢ B ⁡ x ⁡ C = A ⁢ B ⁡ C
8 2 7 sylan9eq ⊢ A ∈ E ∧ B ∈ D ∧ C ∈ ℝ → A ⋅ v B ⁡ C = A ⁢ B ⁡ C
9 8 3impa ⊢ A ∈ E ∧ B ∈ D ∧ C ∈ ℝ → A ⋅ v B ⁡ C = A ⁢ B ⁡ C