Theorem axhv0cl-zf

Description: Derive Axiom ax-hv0cl from Hilbert space under ZF set theory. (Contributed by NM, 31-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	axhv0cl-zf	$$⊢0_{ℎ}\in \mathscr{H}$$

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		df-h0v	$$⊢0_{ℎ}=0_{\mathrm{vec}}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
7	1	fveq2i	$$⊢0_{\mathrm{vec}}\left(U\right)=0_{\mathrm{vec}}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
8	6 7	eqtr4i	$$⊢0_{ℎ}=0_{\mathrm{vec}}\left(U\right)$$
9	5 8	hl0cl	$$⊢U\in {CHil}_{\mathrm{OLD}}\to 0_{ℎ}\in \mathscr{H}$$
10	2 9	ax-mp	$$⊢0_{ℎ}\in \mathscr{H}$$