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Theorem nvdir 30561

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nvdir	⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

**Proof of Theorem nvdir**
Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30545	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		nvdi.2	. . . 4 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	3	vafval 30533	. . 3 ⊢ 𝐺 = (1^st ‘(1^st ‘𝑈))
5		nvdi.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30535	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvdi.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30534	. . 3 ⊢ 𝑋 = ran 𝐺
9	4, 6, 8	vcdir 30496	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
10	2, 9	sylan 578	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 1^st c1st 7993 ℂcc 11147 + caddc 11152 CVec_OLDcvc 30488 NrmCVeccnv 30514 +_𝑣 cpv 30515 BaseSetcba 30516 ·_𝑠OLD cns 30517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-1st 7995 df-2nd 7996 df-vc 30489 df-nv 30522 df-va 30525 df-ba 30526 df-sm 30527 df-0v 30528 df-nmcv 30530

This theorem is referenced by: nvge0 30603 smcnlem 30627 ipidsq 30640 ip2i 30758 ipasslem1 30761 ipasslem11 30770 hldir 30838

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