nvscl - Metamath Proof Explorer

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Theorem nvscl 30573

Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nvscl	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

**Proof of Theorem nvscl**
Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30562	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2734	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30550	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30552	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30551	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vccl 30510	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
10	2, 9	syl3an1 1163	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6541 (class class class)co 7413 1^st c1st 7994 ℂcc 11135 CVec_OLDcvc 30505 NrmCVeccnv 30531 +_𝑣 cpv 30532 BaseSetcba 30533 ·_𝑠OLD cns 30534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7416 df-oprab 7417 df-1st 7996 df-2nd 7997 df-vc 30506 df-nv 30539 df-va 30542 df-ba 30543 df-sm 30544 df-0v 30545 df-nmcv 30547

This theorem is referenced by: nvmval2 30590 nvmf 30592 nvmdi 30595 nvnegneg 30596 nvpncan2 30600 nvaddsub4 30604 nvdif 30613 nvpi 30614 nvmtri 30618 nvabs 30619 nvge0 30620 imsmetlem 30637 smcnlem 30644 ipval2lem2 30651 4ipval2 30655 ipval3 30656 sspmval 30680 lnocoi 30704 lnomul 30707 0lno 30737 nmlno0lem 30740 nmblolbii 30746 blocnilem 30751 ip0i 30772 ip1ilem 30773 ipdirilem 30776 ipasslem1 30778 ipasslem2 30779 ipasslem4 30781 ipasslem5 30782 ipasslem8 30784 ipasslem9 30785 ipasslem10 30786 ipasslem11 30787 dipassr 30793 dipsubdir 30795 siilem1 30798 ipblnfi 30802 ubthlem2 30818 minvecolem2 30822 hhshsslem2 31215

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