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Theorem nvscl 24005
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 2442 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 23992 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2442 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 23980 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 23982 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 23981 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 23927 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
102, 9syl3an1 1251 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 965    = wceq 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090   1stc1st 6574   CCcc 9279   CVecOLDcvc 23922   NrmCVeccnv 23961   +vcpv 23962   BaseSetcba 23963   .sOLDcns 23964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-1st 6576  df-2nd 6577  df-vc 23923  df-nv 23969  df-va 23972  df-ba 23973  df-sm 23974  df-0v 23975  df-nmcv 23977
This theorem is referenced by:  nvmval2  24022  nvzs  24024  nvmf  24025  nvmdi  24029  nvnegneg  24030  nvsubadd  24034  nvpncan2  24035  nvaddsub4  24040  nvnncan  24042  nvdif  24052  nvpi  24053  nvmtri  24058  nvabs  24060  nvge0  24061  imsmetlem  24080  smcnlem  24091  ipval2lem2  24098  4ipval2  24102  ipval3  24103  ipval2lem5  24104  sspmval  24130  sspival  24135  lnocoi  24156  lnomul  24159  0lno  24189  nmlno0lem  24192  nmblolbii  24198  blocnilem  24203  ip0i  24224  ip1ilem  24225  ipdirilem  24228  ipasslem1  24230  ipasslem2  24231  ipasslem4  24233  ipasslem5  24234  ipasslem8  24236  ipasslem9  24237  ipasslem10  24238  ipasslem11  24239  dipassr  24245  dipsubdir  24247  siilem1  24250  ipblnfi  24255  ubthlem2  24271  minvecolem2  24275  hhshsslem2  24668
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