nvscl - Metamath Proof Explorer

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Theorem nvscl 31237

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 2760	. . 3 ⊢ (1^st ‘𝑈) = (1^st ‘𝑈)
2	1	nvvc 31226	. 2 ⊢ (𝑈 ∈ NrmCVec → (1^st ‘𝑈) ∈ CVec_OLD)
3		eqid 2760	. . . 4 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
4	3	vafval 31214	. . 3 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
5		nvscl.4	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
6	5	smfval 31216	. . 3 ⊢ 𝑆 = (2^nd ‘(1^st ‘𝑈))
7		nvscl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
8	7, 3	bafval 31215	. . 3 ⊢ 𝑋 = ran ( +_𝑣 ‘𝑈)
9	4, 6, 8	vccl 31174	. 2 ⊢ (((1^st ‘𝑈) ∈ CVec_OLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
10	2, 9	syl3an1 1181	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ‘cfv 6543 (class class class)co 7424 1^st c1st 8003 ℂcc 11205 CVec_OLDcvc 31169 NrmCVeccnv 31195 +_𝑣 cpv 31196 BaseSetcba 31197 ·_𝑠OLD cns 31198

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pr 5391 ax-un 7755

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7427 df-oprab 7428 df-1st 8005 df-2nd 8006 df-vc 31170 df-nv 31203 df-va 31206 df-ba 31207 df-sm 31208 df-0v 31209 df-nmcv 31211

This theorem is used by: nvmval2 31254 nvmf 31256 nvmdi 31259 nvnegneg 31260 nvpncan2 31264 nvaddsub4 31268 nvdif 31277 nvpi 31278 nvmtri 31282 nvabs 31283 nvge0 31284 imsmetlem 31301 smcnlem 31308 ipval2lem2 31315 4ipval2 31319 ipval3 31320 sspmval 31344 lnocoi 31368 lnomul 31371 0lno 31401 nmlno0lem 31404 nmblolbii 31410 blocnilem 31415 ip0i 31436 ip1ilem 31437 ipdirilem 31440 ipasslem1 31442 ipasslem2 31443 ipasslem4 31445 ipasslem5 31446 ipasslem8 31448 ipasslem9 31449 ipasslem10 31450 ipasslem11 31451 dipassr 31457 dipsubdir 31459 siilem1 31462 ipblnfi 31466 ubthlem2 31482 minvecolem2 31486 hhshsslem2 31879

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