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Theorem axhfvmul-zf 31339

Description: Derive Axiom ax-hfvmul 31357 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 31321	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6884	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2789	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31244	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31327	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulf 31256	. 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 1570 ∈ wcel 2143 ⟨cop 4595 × cxp 5659 ⟶wf 6532 ‘cfv 6536 ℂcc 11093 BaseSetcba 30938 CHil_OLDchlo 31237 ℋchba 31271 +_ℎ cva 31272 ·_ℎ csm 31273 norm_ℎcno 31275

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-1st 7982 df-2nd 7983 df-vc 30911 df-nv 30944 df-va 30947 df-ba 30948 df-sm 30949 df-0v 30950 df-nmcv 30952 df-cbn 31215 df-hlo 31238 df-hba 31321

This theorem is referenced by: (None)

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