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Theorem nvvop 23992
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  =  ( .sOLD `  U )
Assertion
Ref Expression
nvvop  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )

Proof of Theorem nvvop
StepHypRef Expression
1 vcrel 23930 . . 3  |-  Rel  CVecOLD
2 nvss 23976 . . . . 5  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
3 nvvop.1 . . . . . . . 8  |-  W  =  ( 1st `  U
)
4 eqid 2443 . . . . . . . 8  |-  ( normCV `  U )  =  (
normCV
`  U )
53, 4nvop2 23991 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  ( normCV `  U
) >. )
65eleq1d 2509 . . . . . 6  |-  ( U  e.  NrmCVec  ->  ( U  e.  NrmCVec  <->  <. W ,  ( normCV `  U
) >.  e.  NrmCVec ) )
76ibi 241 . . . . 5  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  NrmCVec )
82, 7sseldi 3359 . . . 4  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  ( CVecOLD  X.  _V ) )
9 opelxp1 4877 . . . 4  |-  ( <. W ,  ( normCV `  U
) >.  e.  ( CVecOLD  X.  _V )  ->  W  e.  CVecOLD )
108, 9syl 16 . . 3  |-  ( U  e.  NrmCVec  ->  W  e.  CVecOLD )
11 1st2nd 6625 . . 3  |-  ( ( Rel  CVecOLD  /\  W  e.  CVecOLD )  ->  W  =  <. ( 1st `  W ) ,  ( 2nd `  W )
>. )
121, 10, 11sylancr 663 . 2  |-  ( U  e.  NrmCVec  ->  W  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >. )
13 nvvop.2 . . . . 5  |-  G  =  ( +v `  U
)
1413vafval 23986 . . . 4  |-  G  =  ( 1st `  ( 1st `  U ) )
153fveq2i 5699 . . . 4  |-  ( 1st `  W )  =  ( 1st `  ( 1st `  U ) )
1614, 15eqtr4i 2466 . . 3  |-  G  =  ( 1st `  W
)
17 nvvop.4 . . . . 5  |-  S  =  ( .sOLD `  U )
1817smfval 23988 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
193fveq2i 5699 . . . 4  |-  ( 2nd `  W )  =  ( 2nd `  ( 1st `  U ) )
2018, 19eqtr4i 2466 . . 3  |-  S  =  ( 2nd `  W
)
2116, 20opeq12i 4069 . 2  |-  <. G ,  S >.  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >.
2212, 21syl6eqr 2493 1  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888    X. cxp 4843   Rel wrel 4850   ` cfv 5423   1stc1st 6580   2ndc2nd 6581   CVecOLDcvc 23928   NrmCVeccnv 23967   +vcpv 23968   .sOLDcns 23970   normCVcnmcv 23973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fo 5429  df-fv 5431  df-oprab 6100  df-1st 6582  df-2nd 6583  df-vc 23929  df-nv 23975  df-va 23978  df-sm 23980  df-nmcv 23983
This theorem is referenced by:  nvi  23997  nvvc  23998  nvop  24070
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