Theorem axhis1-zf

Description: Derive Axiom ax-his1 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	axhis1-zf	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{\cdot }_{\mathrm{ih}}B=\overline{B{\cdot }_{\mathrm{ih}}A}$$

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	hlipcj	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{\cdot }_{\mathrm{ih}}B=\overline{B{\cdot }_{\mathrm{ih}}A}$$
8	2 7	mp3an1	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{\cdot }_{\mathrm{ih}}B=\overline{B{\cdot }_{\mathrm{ih}}A}$$