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Theorem hldi 30851

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

**Hypotheses**
Ref	Expression
hldi.1	⊢ 𝑋 = (BaseSet‘𝑈)
hldi.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
hldi.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
hldi	⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

**Proof of Theorem hldi**
Step	Hyp	Ref	Expression
1		hlnv 30835	. 2 ⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ NrmCVec)
2		hldi.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hldi.2	. . 3 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4		hldi.4	. . 3 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
5	2, 3, 4	nvdi 30574	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
6	1, 5	sylan 580	1 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 NrmCVeccnv 30528 +_𝑣 cpv 30529 BaseSetcba 30530 ·_𝑠OLD cns 30531 CHil_OLDchlo 30829

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 5218 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671

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 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-1st 7924 df-2nd 7925 df-vc 30503 df-nv 30536 df-va 30539 df-ba 30540 df-sm 30541 df-0v 30542 df-nmcv 30544 df-cbn 30807 df-hlo 30830

This theorem is referenced by: axhvdistr1-zf 30934

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