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Mirrors > Home > MPE Home > Th. List > nvscl | Structured version Visualization version GIF version |
Description: Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvscl.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvscl.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvscl | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈) | |
2 | 1 | nvvc 28319 | . 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD) |
3 | eqid 2818 | . . . 4 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
4 | 3 | vafval 28307 | . . 3 ⊢ ( +𝑣 ‘𝑈) = (1st ‘(1st ‘𝑈)) |
5 | nvscl.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
6 | 5 | smfval 28309 | . . 3 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
7 | nvscl.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
8 | 7, 3 | bafval 28308 | . . 3 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
9 | 4, 6, 8 | vccl 28267 | . 2 ⊢ (((1st ‘𝑈) ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
10 | 2, 9 | syl3an1 1155 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 ℂcc 10523 CVecOLDcvc 28262 NrmCVeccnv 28288 +𝑣 cpv 28289 BaseSetcba 28290 ·𝑠OLD cns 28291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-1st 7678 df-2nd 7679 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 |
This theorem is referenced by: nvmval2 28347 nvmf 28349 nvmdi 28352 nvnegneg 28353 nvpncan2 28357 nvaddsub4 28361 nvdif 28370 nvpi 28371 nvmtri 28375 nvabs 28376 nvge0 28377 imsmetlem 28394 smcnlem 28401 ipval2lem2 28408 4ipval2 28412 ipval3 28413 sspmval 28437 lnocoi 28461 lnomul 28464 0lno 28494 nmlno0lem 28497 nmblolbii 28503 blocnilem 28508 ip0i 28529 ip1ilem 28530 ipdirilem 28533 ipasslem1 28535 ipasslem2 28536 ipasslem4 28538 ipasslem5 28539 ipasslem8 28541 ipasslem9 28542 ipasslem10 28543 ipasslem11 28544 dipassr 28550 dipsubdir 28552 siilem1 28555 ipblnfi 28559 ubthlem2 28575 minvecolem2 28579 hhshsslem2 28972 |
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