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Theorem nvscl 21184
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  =  ( .s OLD `  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 2283 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21171 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2283 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 21159 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 21161 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 21160 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 21106 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
102, 9syl3an1 1215 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   1stc1st 6120   CCcc 8735   CVec OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143
This theorem is referenced by:  nvmval2  21201  nvzs  21203  nvmf  21204  nvmdi  21208  nvnegneg  21209  nvsubadd  21213  nvpncan2  21214  nvaddsub4  21219  nvnncan  21221  nvdif  21231  nvpi  21232  nvmtri  21237  nvabs  21239  nvge0  21240  imsmetlem  21259  smcnlem  21270  ipval2lem2  21277  4ipval2  21281  ipval3  21282  ipval2lem5  21283  sspmval  21309  sspival  21314  lnocoi  21335  lnomul  21338  0lno  21368  nmlno0lem  21371  nmblolbii  21377  blocnilem  21382  ip0i  21403  ip1ilem  21404  ipdirilem  21407  ipasslem1  21409  ipasslem2  21410  ipasslem4  21412  ipasslem5  21413  ipasslem8  21415  ipasslem9  21416  ipasslem10  21417  ipasslem11  21418  dipassr  21424  dipsubdir  21426  siilem1  21429  ipblnfi  21434  ubthlem2  21450  minvecolem2  21454  hhshsslem2  21845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156
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