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 = ( BaseSet ‘ 𝑈 )
	hlipf.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	hlipf	⊢ ( 𝑈 ∈ CHilOLD → 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ )

Proof

Step	Hyp	Ref	Expression
1		hlipf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlipf.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
3		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
4	1 2	ipf	⊢ ( 𝑈 ∈ NrmCVec → 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ )
5	3 4	syl	⊢ ( 𝑈 ∈ CHilOLD → 𝑃 : ( 𝑋 × 𝑋 ) ⟶ ℂ )