Theorem hlcom

Description: Hilbert space vector addition is commutative. (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	hlcom	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\wedge B\in X\right)\to AGB=BGA$$

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	nvcom	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\wedge B\in X\right)\to AGB=BGA$$
5	3 4	syl3an1	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\wedge B\in X\right)\to AGB=BGA$$