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Theorem axhfvmul-zf 30965

Description: Derive Axiom ax-hfvmul 30983 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
axhil.1	⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
axhil.2	⊢ 𝑈 ∈ CHil_OLD

**Assertion**
Ref	Expression
axhfvmul-zf	⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ

**Proof of Theorem axhfvmul-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 30947	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6825	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2757	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30870	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 30953	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulf 30882	. 2 ⊢ (𝑈 ∈ CHil_OLD → ·_ℎ :(ℂ × ℋ)⟶ ℋ)
9	1, 8	ax-mp 5	1 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∈ wcel 2111 ⟨cop 4582 × cxp 5614 ⟶wf 6477 ‘cfv 6481 ℂcc 11004 BaseSetcba 30564 CHil_OLDchlo 30863 ℋchba 30897 +_ℎ cva 30898 ·_ℎ csm 30899 norm_ℎcno 30901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-1st 7921 df-2nd 7922 df-vc 30537 df-nv 30570 df-va 30573 df-ba 30574 df-sm 30575 df-0v 30576 df-nmcv 30578 df-cbn 30841 df-hlo 30864 df-hba 30947

This theorem is referenced by: (None)

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