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Theorem axhfvmul-zf 31147

Description: Derive Axiom ax-hfvmul 31165 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 31129	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6865	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2787	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31052	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31135	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulf 31064	. 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 1559 ∈ wcel 2141 ⟨cop 4585 × cxp 5641 ⟶wf 6512 ‘cfv 6516 ℂcc 11065 BaseSetcba 30746 CHil_OLDchlo 31045 ℋchba 31079 +_ℎ cva 31080 ·_ℎ csm 31081 norm_ℎcno 31083

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7394 df-oprab 7395 df-1st 7965 df-2nd 7966 df-vc 30719 df-nv 30752 df-va 30755 df-ba 30756 df-sm 30757 df-0v 30758 df-nmcv 30760 df-cbn 31023 df-hlo 31046 df-hba 31129

This theorem is referenced by: (None)

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