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Theorem axhfvmul-zf 30916

Description: Derive Axiom ax-hfvmul 30934 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 30898	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6861	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2755	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 30821	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 30904	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulf 30833	. 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 1540 ∈ wcel 2109 ⟨cop 4595 × cxp 5636 ⟶wf 6507 ‘cfv 6511 ℂcc 11066 BaseSetcba 30515 CHil_OLDchlo 30814 ℋchba 30848 +_ℎ cva 30849 ·_ℎ csm 30850 norm_ℎcno 30852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-1st 7968 df-2nd 7969 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-nmcv 30529 df-cbn 30792 df-hlo 30815 df-hba 30898

This theorem is referenced by: (None)

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