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Theorem nvscl 21200
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 2296 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21187 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2296 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 21175 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 21177 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 21176 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 21122 . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   1stc1st 6136   CCcc 8751   CVec OLDcvc 21117   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159
This theorem is referenced by:  nvmval2  21217  nvzs  21219  nvmf  21220  nvmdi  21224  nvnegneg  21225  nvsubadd  21229  nvpncan2  21230  nvaddsub4  21235  nvnncan  21237  nvdif  21247  nvpi  21248  nvmtri  21253  nvabs  21255  nvge0  21256  imsmetlem  21275  smcnlem  21286  ipval2lem2  21293  4ipval2  21297  ipval3  21298  ipval2lem5  21299  sspmval  21325  sspival  21330  lnocoi  21351  lnomul  21354  0lno  21384  nmlno0lem  21387  nmblolbii  21393  blocnilem  21398  ip0i  21419  ip1ilem  21420  ipdirilem  21423  ipasslem1  21425  ipasslem2  21426  ipasslem4  21428  ipasslem5  21429  ipasslem8  21431  ipasslem9  21432  ipasslem10  21433  ipasslem11  21434  dipassr  21440  dipsubdir  21442  siilem1  21445  ipblnfi  21450  ubthlem2  21466  minvecolem2  21470  hhshsslem2  21861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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