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Theorem nvscl 25816
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 2400 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25803 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2400 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25791 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25793 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25792 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 25738 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
102, 9syl3an1 1261 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 972    = wceq 1403    e. wcel 1840   ` cfv 5523  (class class class)co 6232   1stc1st 6734   CCcc 9438   CVecOLDcvc 25733   NrmCVeccnv 25772   +vcpv 25773   BaseSetcba 25774   .sOLDcns 25775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-1st 6736  df-2nd 6737  df-vc 25734  df-nv 25780  df-va 25783  df-ba 25784  df-sm 25785  df-0v 25786  df-nmcv 25788
This theorem is referenced by:  nvmval2  25833  nvzs  25835  nvmf  25836  nvmdi  25840  nvnegneg  25841  nvsubadd  25845  nvpncan2  25846  nvaddsub4  25851  nvnncan  25853  nvdif  25863  nvpi  25864  nvmtri  25869  nvabs  25871  nvge0  25872  imsmetlem  25891  smcnlem  25902  ipval2lem2  25909  4ipval2  25913  ipval3  25914  ipval2lem5  25915  sspmval  25941  sspival  25946  lnocoi  25967  lnomul  25970  0lno  26000  nmlno0lem  26003  nmblolbii  26009  blocnilem  26014  ip0i  26035  ip1ilem  26036  ipdirilem  26039  ipasslem1  26041  ipasslem2  26042  ipasslem4  26044  ipasslem5  26045  ipasslem8  26047  ipasslem9  26048  ipasslem10  26049  ipasslem11  26050  dipassr  26056  dipsubdir  26058  siilem1  26061  ipblnfi  26066  ubthlem2  26082  minvecolem2  26086  hhshsslem2  26479
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