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Theorem nvscl 25185
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  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )

Proof of Theorem nvscl
StepHypRef Expression
1 eqid 2462 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25172 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2462 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25160 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25162 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25161 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 25107 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
102, 9syl3an1 1256 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277   1stc1st 6774   CCcc 9481   CVecOLDcvc 25102   NrmCVeccnv 25141   +vcpv 25142   BaseSetcba 25143   .sOLDcns 25144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-1st 6776  df-2nd 6777  df-vc 25103  df-nv 25149  df-va 25152  df-ba 25153  df-sm 25154  df-0v 25155  df-nmcv 25157
This theorem is referenced by:  nvmval2  25202  nvzs  25204  nvmf  25205  nvmdi  25209  nvnegneg  25210  nvsubadd  25214  nvpncan2  25215  nvaddsub4  25220  nvnncan  25222  nvdif  25232  nvpi  25233  nvmtri  25238  nvabs  25240  nvge0  25241  imsmetlem  25260  smcnlem  25271  ipval2lem2  25278  4ipval2  25282  ipval3  25283  ipval2lem5  25284  sspmval  25310  sspival  25315  lnocoi  25336  lnomul  25339  0lno  25369  nmlno0lem  25372  nmblolbii  25378  blocnilem  25383  ip0i  25404  ip1ilem  25405  ipdirilem  25408  ipasslem1  25410  ipasslem2  25411  ipasslem4  25413  ipasslem5  25414  ipasslem8  25416  ipasslem9  25417  ipasslem10  25418  ipasslem11  25419  dipassr  25425  dipsubdir  25427  siilem1  25430  ipblnfi  25435  ubthlem2  25451  minvecolem2  25455  hhshsslem2  25848
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