Theorem axhvmul0-zf

Description: Derive axiom ax-hvmul0 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	axhvmul0-zf	$$⊢A\in \mathscr{H}\to 0{\cdot }_{ℎ}A=0_{ℎ}$$

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	2	hlnvi	$$⊢U\in \mathrm{NrmCVec}$$
7	1 6	h2hsm	$$⊢{\cdot }_{ℎ}={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
8		df-h0v	$$⊢0_{ℎ}=0_{\mathrm{vec}}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
9	1	fveq2i	$$⊢0_{\mathrm{vec}}\left(U\right)=0_{\mathrm{vec}}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
10	8 9	eqtr4i	$$⊢0_{ℎ}=0_{\mathrm{vec}}\left(U\right)$$
11	5 7 10	hlmul0	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in \mathscr{H}\right)\to 0{\cdot }_{ℎ}A=0_{ℎ}$$
12	2 11	mpan	$$⊢A\in \mathscr{H}\to 0{\cdot }_{ℎ}A=0_{ℎ}$$