Theorem ipfval

Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	ipffval.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ipffval.2	⊢ , = ( ·𝑖 ‘ 𝑊 )
	ipffval.3	⊢ · = ( ·if ‘ 𝑊 )
Assertion	ipfval	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		ipffval.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ipffval.2	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		ipffval.3	⊢ · = ( ·if ‘ 𝑊 )
4		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 , 𝑦 ) = ( 𝑋 , 𝑌 ) )
5	1 2 3	ipffval	⊢ · = ( 𝑥 ∈ 𝑉 , 𝑦 ∈ 𝑉 ↦ ( 𝑥 , 𝑦 ) )
6		ovex	⊢ ( 𝑋 , 𝑌 ) ∈ V
7	4 5 6	ovmpoa	⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) )