Theorem mulvval

Description: Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012)

	Ref	Expression
Assertion	mulvval	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 .𝑣 𝐵 ) = ( 𝑣 ∈ ℝ ↦ ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
2		elex	⊢ ( 𝐵 ∈ 𝐷 → 𝐵 ∈ V )
3		fveq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 ‘ 𝑣 ) = ( 𝐵 ‘ 𝑣 ) )
4		oveq12	⊢ ( ( 𝑥 = 𝐴 ∧ ( 𝑦 ‘ 𝑣 ) = ( 𝐵 ‘ 𝑣 ) ) → ( 𝑥 · ( 𝑦 ‘ 𝑣 ) ) = ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) )
5	3 4	sylan2	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 · ( 𝑦 ‘ 𝑣 ) ) = ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) )
6	5	mpteq2dv	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑣 ∈ ℝ ↦ ( 𝑥 · ( 𝑦 ‘ 𝑣 ) ) ) = ( 𝑣 ∈ ℝ ↦ ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) ) )
7		df-mulv	⊢ .𝑣 = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ( 𝑣 ∈ ℝ ↦ ( 𝑥 · ( 𝑦 ‘ 𝑣 ) ) ) )
8		reex	⊢ ℝ ∈ V
9	8	mptex	⊢ ( 𝑣 ∈ ℝ ↦ ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) ) ∈ V
10	6 7 9	ovmpoa	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 .𝑣 𝐵 ) = ( 𝑣 ∈ ℝ ↦ ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) ) )
11	1 2 10	syl2an	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 .𝑣 𝐵 ) = ( 𝑣 ∈ ℝ ↦ ( 𝐴 · ( 𝐵 ‘ 𝑣 ) ) ) )