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Theorem nvvc 21171
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 2283 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2283 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
41, 2, 3nvvop 21165 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( +v `  U ) ,  ( .s OLD `  U ) >. )
5 eqid 2283 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  U )
6 eqid 2283 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  U )
7 eqid 2283 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
85, 2, 3, 6, 7nvi 21170 . . 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 967 . 2  |-  ( U  e.  NrmCVec  ->  <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD )
104, 9eqeltrd 2357 1  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    <_ cle 8868   abscabs 11719   CVec
OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s
OLDcns 21143   0veccn0v 21144   normCVcnmcv 21146
This theorem is referenced by:  nvablo  21172  nvsf  21175  nvscl  21184  nvsid  21185  nvsass  21186  nvdi  21188  nvdir  21189  nv2  21190  nv0  21195  nvsz  21196  nvinv  21197  phop  21396  ip0i  21403  ipdirilem  21407  hlvc  21472
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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