Theorem axhfi-zf

Description: Derive Axiom ax-hfi 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	$$⊢U\in {CHil}_{\mathrm{OLD}}$$
	axhfi.1	$$⊢{\cdot }_{\mathrm{ih}}={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
Assertion	axhfi-zf	$$⊢{\cdot }_{\mathrm{ih}}:\mathscr{H}\times \mathscr{H}⟶ℂ$$

Proof

Step	Hyp	Ref	Expression
1		axhil.1	$$⊢U=⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩$$
2		axhil.2	$$⊢U\in {CHil}_{\mathrm{OLD}}$$
3		axhfi.1	$$⊢{\cdot }_{\mathrm{ih}}={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
4		df-hba	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
5	1	fveq2i	$$⊢\mathrm{BaseSet}\left(U\right)=\mathrm{BaseSet}\left(⟨⟨{+}_{ℎ},{\cdot }_{ℎ}⟩,{norm}_{ℎ}⟩\right)$$
6	4 5	eqtr4i	$$⊢\mathscr{H}=\mathrm{BaseSet}\left(U\right)$$
7	6 3	hlipf	$$⊢U\in {CHil}_{\mathrm{OLD}}\to {\cdot }_{\mathrm{ih}}:\mathscr{H}\times \mathscr{H}⟶ℂ$$
8	2 7	ax-mp	$$⊢{\cdot }_{\mathrm{ih}}:\mathscr{H}\times \mathscr{H}⟶ℂ$$