Theorem hlmulf

Description: Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlmulf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlmulf.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
Assertion	hlmulf	⊢ ( 𝑈 ∈ CHilOLD → 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		hlmulf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlmulf.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
4	1 2	nvsf	⊢ ( 𝑈 ∈ NrmCVec → 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 )
5	3 4	syl	⊢ ( 𝑈 ∈ CHilOLD → 𝑆 : ( ℂ × 𝑋 ) ⟶ 𝑋 )