Theorem axhis3-zf

Description: Derive Axiom ax-his3 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	axhis3-zf	$$⊢\left(A\in ℂ\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{\cdot }_{ℎ}B\right){\cdot }_{\mathrm{ih}}C=A\left(B{\cdot }_{\mathrm{ih}}C\right)$$

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	2	hlnvi	$$⊢U\in \mathrm{NrmCVec}$$
8	1 7	h2hsm	$$⊢{\cdot }_{ℎ}={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
9	6 8 3	hlipass	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\right)\to \left(A{\cdot }_{ℎ}B\right){\cdot }_{\mathrm{ih}}C=A\left(B{\cdot }_{\mathrm{ih}}C\right)$$
10	2 9	mpan	$$⊢\left(A\in ℂ\wedge B\in \mathscr{H}\wedge C\in \mathscr{H}\right)\to \left(A{\cdot }_{ℎ}B\right){\cdot }_{\mathrm{ih}}C=A\left(B{\cdot }_{\mathrm{ih}}C\right)$$