Theorem axhvcom-zf

Description: Derive Axiom ax-hvcom from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	axhil.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
	axhil.2	$$⊢U\in {CHil}_{\mathrm{OLD}}$$
Assertion	axhvcom-zf	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{+}_{ℎ}B=B{+}_{ℎ}A$$

Proof

Step	Hyp	Ref	Expression
1		axhil.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		axhil.2	$$⊢U\in {CHil}_{\mathrm{OLD}}$$
3		df-hba	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
4	1	fveq2i	$$⊢\mathrm{BaseSet}\left(U\right)=\mathrm{BaseSet}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
5	3 4	eqtr4i	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
6	2	hlnvi	$$⊢U\in \mathrm{NrmCVec}$$
7	1 6	h2hva	$$⊢{+}_{ℎ}={+}_v\left(U\right)$$
8	5 7	hlcom	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{+}_{ℎ}B=B{+}_{ℎ}A$$
9	2 8	mp3an1	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{+}_{ℎ}B=B{+}_{ℎ}A$$