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Theorem hldi 30966

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 30950	. 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 30689	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
6	1, 5	sylan 581	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 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 NrmCVeccnv 30643 +_𝑣 cpv 30644 BaseSetcba 30645 ·_𝑠OLD cns 30646 CHil_OLDchlo 30944

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pr 5364 ax-un 7678

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-ov 7359 df-oprab 7360 df-1st 7931 df-2nd 7932 df-vc 30618 df-nv 30651 df-va 30654 df-ba 30655 df-sm 30656 df-0v 30657 df-nmcv 30659 df-cbn 30922 df-hlo 30945

This theorem is referenced by: axhvdistr1-zf 31049

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