nvscl - Metamath Proof Explorer

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Theorem nvscl 30714

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 2737	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 30703	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2737	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 30691	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 30693	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 30692	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vccl 30651	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
10	2, 9	syl3an1 1164	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 ℂcc 11036 CVec_OLDcvc 30646 NrmCVeccnv 30672 +_𝑣 cpv 30673 BaseSetcba 30674 ·_𝑠OLD cns 30675

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 30647 df-nv 30680 df-va 30683 df-ba 30684 df-sm 30685 df-0v 30686 df-nmcv 30688

This theorem is referenced by: nvmval2 30731 nvmf 30733 nvmdi 30736 nvnegneg 30737 nvpncan2 30741 nvaddsub4 30745 nvdif 30754 nvpi 30755 nvmtri 30759 nvabs 30760 nvge0 30761 imsmetlem 30778 smcnlem 30785 ipval2lem2 30792 4ipval2 30796 ipval3 30797 sspmval 30821 lnocoi 30845 lnomul 30848 0lno 30878 nmlno0lem 30881 nmblolbii 30887 blocnilem 30892 ip0i 30913 ip1ilem 30914 ipdirilem 30917 ipasslem1 30919 ipasslem2 30920 ipasslem4 30922 ipasslem5 30923 ipasslem8 30925 ipasslem9 30926 ipasslem10 30927 ipasslem11 30928 dipassr 30934 dipsubdir 30936 siilem1 30939 ipblnfi 30943 ubthlem2 30959 minvecolem2 30963 hhshsslem2 31356

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