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Theorem nvvop 21165
Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvvop.1  |-  W  =  ( 1st `  U
)
nvvop.2  |-  G  =  ( +v `  U
)
nvvop.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nvvop  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )

Proof of Theorem nvvop
StepHypRef Expression
1 vcrel 21103 . . 3  |-  Rel  CVec OLD
2 nvss 21149 . . . . 5  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
3 nvvop.1 . . . . . . . 8  |-  W  =  ( 1st `  U
)
4 eqid 2283 . . . . . . . 8  |-  ( normCV `  U )  =  (
normCV
`  U )
53, 4nvop2 21164 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  ( normCV `  U
) >. )
65eleq1d 2349 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( U  e.  NrmCVec  <->  <. W ,  ( normCV `  U
) >.  e.  NrmCVec ) )
76ibi 232 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  NrmCVec )
82, 7sseldi 3178 . . . 4  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  ( CVec OLD  X.  _V ) )
9 opelxp1 4722 . . . 4  |-  ( <. W ,  ( normCV `  U
) >.  e.  ( CVec
OLD  X.  _V )  ->  W  e.  CVec OLD )
108, 9syl 15 . . 3  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
11 1st2nd 6166 . . 3  |-  ( ( Rel  CVec OLD  /\  W  e. 
CVec OLD )  ->  W  =  <. ( 1st `  W
) ,  ( 2nd `  W ) >. )
121, 10, 11sylancr 644 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >. )
13 nvvop.2 . . . . 5  |-  G  =  ( +v `  U
)
1413vafval 21159 . . . 4  |-  G  =  ( 1st `  ( 1st `  U ) )
153fveq2i 5528 . . . 4  |-  ( 1st `  W )  =  ( 1st `  ( 1st `  U ) )
1614, 15eqtr4i 2306 . . 3  |-  G  =  ( 1st `  W
)
17 nvvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
1817smfval 21161 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
193fveq2i 5528 . . . 4  |-  ( 2nd `  W )  =  ( 2nd `  ( 1st `  U ) )
2018, 19eqtr4i 2306 . . 3  |-  S  =  ( 2nd `  W
)
2116, 20opeq12i 3801 . 2  |-  <. G ,  S >.  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >.
2212, 21syl6eqr 2333 1  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   CVec OLDcvc 21101   NrmCVeccnv 21140   +vcpv 21141   .s OLDcns 21143   normCVcnmcv 21146
This theorem is referenced by:  nvi  21170  nvvc  21171  nvop  21243
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-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-rab 2552  df-v 2790  df-sbc 2992  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-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-fo 5261  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-sm 21153  df-nmcv 21156
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