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Theorem phop 25564
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
phop.2  |-  G  =  ( +v `  U
)
phop.4  |-  S  =  ( .sOLD `  U )
phop.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
phop  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )

Proof of Theorem phop
StepHypRef Expression
1 phrel 25561 . . 3  |-  Rel  CPreHil OLD
2 1st2nd 6841 . . 3  |-  ( ( Rel  CPreHil OLD  /\  U  e.  CPreHil
OLD )  ->  U  =  <. ( 1st `  U
) ,  ( 2nd `  U ) >. )
31, 2mpan 670 . 2  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. ( 1st `  U
) ,  ( 2nd `  U ) >. )
4 phop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 25331 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 4223 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 phnv 25560 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
8 eqid 2467 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
98nvvc 25339 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
10 vcrel 25271 . . . . . . 7  |-  Rel  CVecOLD
11 1st2nd 6841 . . . . . . 7  |-  ( ( Rel  CVecOLD  /\  ( 1st `  U )  e. 
CVecOLD )  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
1210, 11mpan 670 . . . . . 6  |-  ( ( 1st `  U )  e.  CVecOLD  ->  ( 1st `  U )  = 
<. ( 1st `  ( 1st `  U ) ) ,  ( 2nd `  ( 1st `  U ) )
>. )
13 phop.2 . . . . . . . 8  |-  G  =  ( +v `  U
)
1413vafval 25327 . . . . . . 7  |-  G  =  ( 1st `  ( 1st `  U ) )
15 phop.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
1615smfval 25329 . . . . . . 7  |-  S  =  ( 2nd `  ( 1st `  U ) )
1714, 16opeq12i 4224 . . . . . 6  |-  <. G ,  S >.  =  <. ( 1st `  ( 1st `  U
) ) ,  ( 2nd `  ( 1st `  U ) ) >.
1812, 17syl6eqr 2526 . . . . 5  |-  ( ( 1st `  U )  e.  CVecOLD  ->  ( 1st `  U )  = 
<. G ,  S >. )
197, 9, 183syl 20 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
2019opeq1d 4225 . . 3  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  N >.  =  <. <. G ,  S >. ,  N >. )
216, 20syl5eqr 2522 . 2  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  ( 2nd `  U
) >.  =  <. <. G ,  S >. ,  N >. )
223, 21eqtrd 2508 1  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   <.cop 4039   Rel wrel 5010   ` cfv 5594   1stc1st 6793   2ndc2nd 6794   CVecOLDcvc 25269   NrmCVeccnv 25308   +vcpv 25309   .sOLDcns 25311   normCVcnmcv 25314   CPreHil OLDccphlo 25558
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-1st 6795  df-2nd 6796  df-vc 25270  df-nv 25316  df-va 25319  df-ba 25320  df-sm 25321  df-0v 25322  df-nmcv 25324  df-ph 25559
This theorem is referenced by:  phpar  25570
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