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Theorem nvscl 28330
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 2818 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28319 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2818 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28307 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28309 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28308 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vccl 28267 . 2 (((1st𝑈) ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
102, 9syl3an1 1155 1 ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (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  cc 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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