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Theorem nvscl 30655
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 2735 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 30644 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2735 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 30632 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 30634 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 30633 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vccl 30592 . 2 (((1st𝑈) ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
102, 9syl3an1 1162 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  1st c1st 8011  cc 11151  CVecOLDcvc 30587  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629
This theorem is referenced by:  nvmval2  30672  nvmf  30674  nvmdi  30677  nvnegneg  30678  nvpncan2  30682  nvaddsub4  30686  nvdif  30695  nvpi  30696  nvmtri  30700  nvabs  30701  nvge0  30702  imsmetlem  30719  smcnlem  30726  ipval2lem2  30733  4ipval2  30737  ipval3  30738  sspmval  30762  lnocoi  30786  lnomul  30789  0lno  30819  nmlno0lem  30822  nmblolbii  30828  blocnilem  30833  ip0i  30854  ip1ilem  30855  ipdirilem  30858  ipasslem1  30860  ipasslem2  30861  ipasslem4  30863  ipasslem5  30864  ipasslem8  30866  ipasslem9  30867  ipasslem10  30868  ipasslem11  30869  dipassr  30875  dipsubdir  30877  siilem1  30880  ipblnfi  30884  ubthlem2  30900  minvecolem2  30904  hhshsslem2  31297
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