Theorem hlipf

Description: Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlipf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlipf.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
Assertion	hlipf	$$⊢U\in {CHil}_{\mathrm{OLD}}\to P:X\times X⟶ℂ$$

Proof

Step	Hyp	Ref	Expression
1		hlipf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hlipf.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
3		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
4	1 2	ipf	$$⊢U\in \mathrm{NrmCVec}\to P:X\times X⟶ℂ$$
5	3 4	syl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to P:X\times X⟶ℂ$$