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Theorem nvscl 28573
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 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvscl ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Proof of Theorem nvscl
StepHypRef Expression
1 eqid 2739 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28562 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2739 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28550 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28552 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28551 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vccl 28510 . 2 (((1st𝑈) ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
102, 9syl3an1 1164 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2114  cfv 6349  (class class class)co 7182  1st c1st 7724  cc 10625  CVecOLDcvc 28505  NrmCVeccnv 28531   +𝑣 cpv 28532  BaseSetcba 28533   ·𝑠OLD cns 28534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7185  df-oprab 7186  df-1st 7726  df-2nd 7727  df-vc 28506  df-nv 28539  df-va 28542  df-ba 28543  df-sm 28544  df-0v 28545  df-nmcv 28547
This theorem is referenced by:  nvmval2  28590  nvmf  28592  nvmdi  28595  nvnegneg  28596  nvpncan2  28600  nvaddsub4  28604  nvdif  28613  nvpi  28614  nvmtri  28618  nvabs  28619  nvge0  28620  imsmetlem  28637  smcnlem  28644  ipval2lem2  28651  4ipval2  28655  ipval3  28656  sspmval  28680  lnocoi  28704  lnomul  28707  0lno  28737  nmlno0lem  28740  nmblolbii  28746  blocnilem  28751  ip0i  28772  ip1ilem  28773  ipdirilem  28776  ipasslem1  28778  ipasslem2  28779  ipasslem4  28781  ipasslem5  28782  ipasslem8  28784  ipasslem9  28785  ipasslem10  28786  ipasslem11  28787  dipassr  28793  dipsubdir  28795  siilem1  28798  ipblnfi  28802  ubthlem2  28818  minvecolem2  28822  hhshsslem2  29215
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