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Theorem hlmulass 30993

Description: Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hlmulf.1	⊢ 𝑋 = (BaseSet‘𝑈)
hlmulf.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
hlmulass	⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

**Proof of Theorem hlmulass**
Step	Hyp	Ref	Expression
1		hlnv 30978	. 2 ⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ NrmCVec)
2		hlmulf.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlmulf.4	. . 3 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
4	2, 3	nvsass 30715	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
5	1, 4	sylan 581	1 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 · cmul 11043 NrmCVeccnv 30671 BaseSetcba 30673 ·_𝑠OLD cns 30674 CHil_OLDchlo 30972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-1st 7943 df-2nd 7944 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-nmcv 30687 df-cbn 30950 df-hlo 30973

This theorem is referenced by: axhvmulass-zf 31076

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