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Theorem axhvmulass-zf 31278

Description: Derive Axiom ax-hvmulass 31296 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
axhvmulass-zf	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))

**Proof of Theorem axhvmulass-zf**
Step	Hyp	Ref	Expression
1		axhil.2	. 2 ⊢ 𝑈 ∈ CHil_OLD
2		df-hba 31258	. . . 4 ⊢ ℋ = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
3		axhil.1	. . . . 5 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
4	3	fveq2i 6882	. . . 4 ⊢ (BaseSet‘𝑈) = (BaseSet‘⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩)
5	2, 4	eqtr4i 2795	. . 3 ⊢ ℋ = (BaseSet‘𝑈)
6	1	hlnvi 31181	. . . 4 ⊢ 𝑈 ∈ NrmCVec
7	3, 6	h2hsm 31264	. . 3 ⊢ ·_ℎ = ( ·_𝑠OLD ‘𝑈)
8	5, 7	hlmulass 31195	. 2 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))
9	1, 8	mpan 702	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·_ℎ 𝐶) = (𝐴 ·_ℎ (𝐵 ·_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ⟨cop 4597 ‘cfv 6534 (class class class)co 7408 ℂcc 11094 · cmul 11101 BaseSetcba 30875 CHil_OLDchlo 31174 ℋchba 31208 +_ℎ cva 31209 ·_ℎ csm 31210 norm_ℎcno 31212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pr 5402 ax-un 7730

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7411 df-oprab 7412 df-1st 7982 df-2nd 7983 df-vc 30848 df-nv 30881 df-va 30884 df-ba 30885 df-sm 30886 df-0v 30887 df-nmcv 30889 df-cbn 31152 df-hlo 31175 df-hba 31258

This theorem is referenced by: (None)

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