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Theorem nvscl 22109
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 2438 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 22096 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 eqid 2438 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 22084 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 22086 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 22085 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vccl 22031 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  CC  /\  B  e.  X )  ->  ( A S B )  e.  X )
102, 9syl3an1 1218 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 937    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   1stc1st 6349   CCcc 8990   CVec OLDcvc 22026   NrmCVeccnv 22065   +vcpv 22066   BaseSetcba 22067   .s
OLDcns 22068
This theorem is referenced by:  nvmval2  22126  nvzs  22128  nvmf  22129  nvmdi  22133  nvnegneg  22134  nvsubadd  22138  nvpncan2  22139  nvaddsub4  22144  nvnncan  22146  nvdif  22156  nvpi  22157  nvmtri  22162  nvabs  22164  nvge0  22165  imsmetlem  22184  smcnlem  22195  ipval2lem2  22202  4ipval2  22206  ipval3  22207  ipval2lem5  22208  sspmval  22234  sspival  22239  lnocoi  22260  lnomul  22263  0lno  22293  nmlno0lem  22296  nmblolbii  22302  blocnilem  22307  ip0i  22328  ip1ilem  22329  ipdirilem  22332  ipasslem1  22334  ipasslem2  22335  ipasslem4  22337  ipasslem5  22338  ipasslem8  22340  ipasslem9  22341  ipasslem10  22342  ipasslem11  22343  dipassr  22349  dipsubdir  22351  siilem1  22354  ipblnfi  22359  ubthlem2  22375  minvecolem2  22379  hhshsslem2  22770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-1st 6351  df-2nd 6352  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-nmcv 22081
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