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Theorem nvvop 21181
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 21119 . . 3  |-  Rel  CVec OLD
2 nvss 21165 . . . . 5  |-  NrmCVec  C_  ( CVec OLD  X.  _V )
3 nvvop.1 . . . . . . . 8  |-  W  =  ( 1st `  U
)
4 eqid 2296 . . . . . . . 8  |-  ( normCV `  U )  =  (
normCV
`  U )
53, 4nvop2 21180 . . . . . . 7  |-  ( U  e.  NrmCVec  ->  U  =  <. W ,  ( normCV `  U
) >. )
65eleq1d 2362 . . . . . 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 3191 . . . 4  |-  ( U  e.  NrmCVec  ->  <. W ,  (
normCV
`  U ) >.  e.  ( CVec OLD  X.  _V ) )
9 opelxp1 4738 . . . 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 6182 . . 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 21175 . . . 4  |-  G  =  ( 1st `  ( 1st `  U ) )
153fveq2i 5544 . . . 4  |-  ( 1st `  W )  =  ( 1st `  ( 1st `  U ) )
1614, 15eqtr4i 2319 . . 3  |-  G  =  ( 1st `  W
)
17 nvvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
1817smfval 21177 . . . 4  |-  S  =  ( 2nd `  ( 1st `  U ) )
193fveq2i 5544 . . . 4  |-  ( 2nd `  W )  =  ( 2nd `  ( 1st `  U ) )
2018, 19eqtr4i 2319 . . 3  |-  S  =  ( 2nd `  W
)
2116, 20opeq12i 3817 . 2  |-  <. G ,  S >.  =  <. ( 1st `  W ) ,  ( 2nd `  W
) >.
2212, 21syl6eqr 2346 1  |-  ( U  e.  NrmCVec  ->  W  =  <. G ,  S >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   CVec OLDcvc 21117   NrmCVeccnv 21156   +vcpv 21157   .s OLDcns 21159   normCVcnmcv 21162
This theorem is referenced by:  nvi  21186  nvvc  21187  nvop  21259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-sm 21169  df-nmcv 21172
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