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Theorem nvop 22159
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop.2  |-  G  =  ( +v `  U
)
nvop.4  |-  S  =  ( .s OLD `  U
)
nvop.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvop  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 22074 . . 3  |-  Rel  NrmCVec
2 1st2nd 6386 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 652 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 22079 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 3981 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 eqid 2436 . . . . 5  |-  ( 1st `  U )  =  ( 1st `  U )
8 nvop.2 . . . . 5  |-  G  =  ( +v `  U
)
9 nvop.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
107, 8, 9nvvop 22081 . . . 4  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
1110opeq1d 3983 . . 3  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
126, 11syl5eqr 2482 . 2  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  =  <. <. G ,  S >. ,  N >. )
133, 12eqtrd 2468 1  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3810   Rel wrel 4876   ` cfv 5447   1stc1st 6340   2ndc2nd 6341   NrmCVeccnv 22056   +vcpv 22057   .s OLDcns 22059   normCVcnmcv 22062
This theorem is referenced by:  sspval  22215  isph  22316  hilhhi  22659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-oprab 6078  df-1st 6342  df-2nd 6343  df-vc 22018  df-nv 22064  df-va 22067  df-sm 22069  df-nmcv 22072
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