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Theorem nvvc 22095
Description: The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvvc.1  |-  W  =  ( 1st `  U
)
Assertion
Ref Expression
nvvc  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )

Proof of Theorem nvvc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvvc.1 . . 3  |-  W  =  ( 1st `  U
)
2 eqid 2437 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2437 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
41, 2, 3nvvop 22089 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( +v `  U ) ,  ( .s OLD `  U ) >. )
5 eqid 2437 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
6 eqid 2437 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
7 eqid 2437 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
85, 2, 3, 6, 7nvi 22094 . . 3  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD  /\  ( normCV `  U ) : ( BaseSet `  U ) --> RR  /\  A. x  e.  ( BaseSet `  U )
( ( ( (
normCV
`  U ) `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( ( normCV `  U ) `  (
y ( .s OLD `  U ) x ) )  =  ( ( abs `  y )  x.  ( ( normCV `  U ) `  x
) )  /\  A. y  e.  ( BaseSet `  U ) ( (
normCV
`  U ) `  ( x ( +v
`  U ) y ) )  <_  (
( ( normCV `  U
) `  x )  +  ( ( normCV `  U ) `  y
) ) ) ) )
98simp1d 970 . 2  |-  ( U  e.  NrmCVec  ->  <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD )
104, 9eqeltrd 2511 1  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   <.cop 3818   class class class wbr 4213   -->wf 5451   ` cfv 5455  (class class class)co 6082   1stc1st 6348   CCcc 8989   RRcr 8990   0cc0 8991    + caddc 8994    x. cmul 8996    <_ cle 9122   abscabs 12040   CVec
OLDcvc 22025   NrmCVeccnv 22064   +vcpv 22065   BaseSetcba 22066   .s
OLDcns 22067   0veccn0v 22068   normCVcnmcv 22070
This theorem is referenced by:  nvablo  22096  nvsf  22099  nvscl  22108  nvsid  22109  nvsass  22110  nvdi  22112  nvdir  22113  nv2  22114  nv0  22119  nvsz  22120  nvinv  22121  phop  22320  ip0i  22327  ipdirilem  22331  hlvc  22396
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-1st 6350  df-2nd 6351  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-nmcv 22080
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