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 ) ) )