Theorem scafval

Description: The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	scaffval.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	scaffval.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	scaffval.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
	scaffval.a	⊢ ∙ = ( ·sf ‘ 𝑊 )
	scaffval.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
Assertion	scafval	⊢ ( ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∙ 𝑌 ) = ( 𝑋 · 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		scaffval.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
2		scaffval.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3		scaffval.k	⊢ 𝐾 = ( Base ‘ 𝐹 )
4		scaffval.a	⊢ ∙ = ( ·sf ‘ 𝑊 )
5		scaffval.s	⊢ · = ( ·𝑠 ‘ 𝑊 )
6		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 · 𝑦 ) = ( 𝑋 · 𝑌 ) )
7	1 2 3 4 5	scaffval	⊢ ∙ = ( 𝑥 ∈ 𝐾 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 · 𝑦 ) )
8		ovex	⊢ ( 𝑋 · 𝑌 ) ∈ V
9	6 7 8	ovmpoa	⊢ ( ( 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∙ 𝑌 ) = ( 𝑋 · 𝑌 ) )