Theorem ipfeq

Description: If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015)

	Ref	Expression
Hypotheses	ipffval.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ipffval.2	⊢ , = ( ·𝑖 ‘ 𝑊 )
	ipffval.3	⊢ · = ( ·if ‘ 𝑊 )
Assertion	ipfeq	⊢ ( , Fn ( 𝑉 × 𝑉 ) → · = , )

Proof

Step	Hyp	Ref	Expression
1		ipffval.1	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ipffval.2	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		ipffval.3	⊢ · = ( ·if ‘ 𝑊 )
4	1 2 3	ipffval	⊢ · = ( 𝑥 ∈ 𝑉 , 𝑦 ∈ 𝑉 ↦ ( 𝑥 , 𝑦 ) )
5		fnov	⊢ ( , Fn ( 𝑉 × 𝑉 ) ↔ , = ( 𝑥 ∈ 𝑉 , 𝑦 ∈ 𝑉 ↦ ( 𝑥 , 𝑦 ) ) )
6	5	biimpi	⊢ ( , Fn ( 𝑉 × 𝑉 ) → , = ( 𝑥 ∈ 𝑉 , 𝑦 ∈ 𝑉 ↦ ( 𝑥 , 𝑦 ) ) )
7	4 6	eqtr4id	⊢ ( , Fn ( 𝑉 × 𝑉 ) → · = , )