Theorem hlmulf

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

	Ref	Expression
Hypotheses	hlmulf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlmulf.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
Assertion	hlmulf	$$⊢U\in {CHil}_{\mathrm{OLD}}\to S:ℂ\times X⟶X$$

Proof

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