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	$$⊢V={\mathrm{Base}}_W$$
	ipffval.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(W\right)$$
	ipffval.3	$$⊢\underset{\dot{}}{·}={\cdot }_{\mathrm{if}}\left(W\right)$$
Assertion	ipfeq	$$⊢\underset{\dot{}}{,}Fn\left(V\times V\right)\to \underset{\dot{}}{·}=\underset{\dot{}}{,}$$

Proof

Step	Hyp	Ref	Expression
1		ipffval.1	$$⊢V={\mathrm{Base}}_W$$
2		ipffval.2	$$⊢\underset{\dot{}}{,}={\cdot }_{𝑖}\left(W\right)$$
3		ipffval.3	$$⊢\underset{\dot{}}{·}={\cdot }_{\mathrm{if}}\left(W\right)$$
4	1 2 3	ipffval	$$⊢\underset{\dot{}}{·}=\left(x\in V,y\in V⟼x\underset{\dot{}}{,}y\right)$$
5		fnov	$$⊢\underset{\dot{}}{,}Fn\left(V\times V\right)\leftrightarrow \underset{\dot{}}{,}=\left(x\in V,y\in V⟼x\underset{\dot{}}{,}y\right)$$
6	5	biimpi	$$⊢\underset{\dot{}}{,}Fn\left(V\times V\right)\to \underset{\dot{}}{,}=\left(x\in V,y\in V⟼x\underset{\dot{}}{,}y\right)$$
7	4 6	eqtr4id	$$⊢\underset{\dot{}}{,}Fn\left(V\times V\right)\to \underset{\dot{}}{·}=\underset{\dot{}}{,}$$