Theorem hladdf

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

	Ref	Expression
Hypotheses	hladdf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hladdf.2	$$⊢G={+}_v\left(U\right)$$
Assertion	hladdf	$$⊢U\in {CHil}_{\mathrm{OLD}}\to G:X\times X⟶X$$

Proof

Step	Hyp	Ref	Expression
1		hladdf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hladdf.2	$$⊢G={+}_v\left(U\right)$$
3		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
4	1 2	nvgf	$$⊢U\in \mathrm{NrmCVec}\to G:X\times X⟶X$$
5	3 4	syl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to G:X\times X⟶X$$