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Theorem nvvop 22090
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 22028 . . 3  |-  Rel  CVec OLD
2 nvss 22074 . . . . 5  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
3 nvvop.1 . . . . . . . 8  |-  W  =  ( 1st `  U
)
4 eqid 2438 . . . . . . . 8  |-  ( normCV `  U )  =  (
normCV
`  U )
53, 4nvop2 22089 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  ( normCV `  U
) >. )
65eleq1d 2504 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( U  e.  NrmCVec  <->  <. W ,  ( normCV `  U
) >.  e.  NrmCVec ) )
76ibi 234 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  NrmCVec )
82, 7sseldi 3348 . . . 4  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  ( CVec OLD  X.  _V ) )
9 opelxp1 4913 . . . 4  |-  ( <. W ,  ( normCV `  U
) >.  e.  ( CVec
OLD  X.  _V )  ->  W  e.  CVec OLD )
108, 9syl 16 . . 3  |-  ( U  e.  NrmCVec  ->  W  e.  CVec OLD )
11 1st2nd 6395 . . 3  |-  ( ( Rel  CVec OLD  /\  W  e. 
CVec OLD )  ->  W  =  <. ( 1st `  W
) ,  ( 2nd `  W ) >. )
121, 10, 11sylancr 646 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >. )
13 nvvop.2 . . . . 5  |-  G  =  ( +v `  U
)
1413vafval 22084 . . . 4  |-  G  =  ( 1st `  ( 1st `  U ) )
153fveq2i 5733 . . . 4  |-  ( 1st `  W )  =  ( 1st `  ( 1st `  U ) )
1614, 15eqtr4i 2461 . . 3  |-  G  =  ( 1st `  W
)
17 nvvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
1817smfval 22086 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
193fveq2i 5733 . . . 4  |-  ( 2nd `  W )  =  ( 2nd `  ( 1st `  U ) )
2018, 19eqtr4i 2461 . . 3  |-  S  =  ( 2nd `  W
)
2116, 20opeq12i 3991 . 2  |-  <. G ,  S >.  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >.
2212, 21syl6eqr 2488 1  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    X. cxp 4878   Rel wrel 4885   ` cfv 5456   1stc1st 6349   2ndc2nd 6350   CVec OLDcvc 22026   NrmCVeccnv 22065   +vcpv 22066   .s OLDcns 22068   normCVcnmcv 22071
This theorem is referenced by:  nvi  22095  nvvc  22096  nvop  22168
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-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-rab 2716  df-v 2960  df-sbc 3164  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-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-fo 5462  df-fv 5464  df-oprab 6087  df-1st 6351  df-2nd 6352  df-vc 22027  df-nv 22073  df-va 22076  df-sm 22078  df-nmcv 22081
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