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Theorem nvvop 25293
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 25231 . . 3  |-  Rel  CVecOLD
2 nvss 25277 . . . . 5  |-  NrmCVec  C_  ( CVecOLD  X.  _V )
3 nvvop.1 . . . . . . . 8  |-  W  =  ( 1st `  U
)
4 eqid 2467 . . . . . . . 8  |-  ( normCV `  U )  =  (
normCV
`  U )
53, 4nvop2 25292 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  ( normCV `  U
) >. )
65eleq1d 2536 . . . . . 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 3507 . . . 4  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  ( CVecOLD  X.  _V ) )
9 opelxp1 5037 . . . 4  |-  ( <. W ,  ( normCV `  U
) >.  e.  ( CVecOLD  X.  _V )  ->  W  e.  CVecOLD )
108, 9syl 16 . . 3  |-  ( U  e.  NrmCVec  ->  W  e.  CVecOLD )
11 1st2nd 6840 . . 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 25287 . . . 4  |-  G  =  ( 1st `  ( 1st `  U ) )
153fveq2i 5874 . . . 4  |-  ( 1st `  W )  =  ( 1st `  ( 1st `  U ) )
1614, 15eqtr4i 2499 . . 3  |-  G  =  ( 1st `  W
)
17 nvvop.4 . . . . 5  |-  S  =  ( .sOLD `  U )
1817smfval 25289 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
193fveq2i 5874 . . . 4  |-  ( 2nd `  W )  =  ( 2nd `  ( 1st `  U ) )
2018, 19eqtr4i 2499 . . 3  |-  S  =  ( 2nd `  W
)
2116, 20opeq12i 4223 . 2  |-  <. G ,  S >.  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >.
2212, 21syl6eqr 2526 1  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118   <.cop 4038    X. cxp 5002   Rel wrel 5009   ` cfv 5593   1stc1st 6792   2ndc2nd 6793   CVecOLDcvc 25229   NrmCVeccnv 25268   +vcpv 25269   .sOLDcns 25271   normCVcnmcv 25274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-oprab 6298  df-1st 6794  df-2nd 6795  df-vc 25230  df-nv 25276  df-va 25279  df-sm 25281  df-nmcv 25284
This theorem is referenced by:  nvi  25298  nvvc  25299  nvop  25371
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