Metamath Proof Explorer


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 , = ( ·𝑖𝑊 )
ipffval.3 · = ( ·if𝑊 )
Assertion ipfval ( ( 𝑋𝑉𝑌𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) )

Proof

Step Hyp Ref Expression
1 ipffval.1 𝑉 = ( Base ‘ 𝑊 )
2 ipffval.2 , = ( ·𝑖𝑊 )
3 ipffval.3 · = ( ·if𝑊 )
4 oveq12 ( ( 𝑥 = 𝑋𝑦 = 𝑌 ) → ( 𝑥 , 𝑦 ) = ( 𝑋 , 𝑌 ) )
5 1 2 3 ipffval · = ( 𝑥𝑉 , 𝑦𝑉 ↦ ( 𝑥 , 𝑦 ) )
6 ovex ( 𝑋 , 𝑌 ) ∈ V
7 4 5 6 ovmpoa ( ( 𝑋𝑉𝑌𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) )