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Theorem phop 24218
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 24215 . . 3  |-  Rel  CPreHil OLD
2 1st2nd 6620 . . 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 23985 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 4063 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 phnv 24214 . . . . 5  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
8 eqid 2443 . . . . . 6  |-  ( 1st `  U )  =  ( 1st `  U )
98nvvc 23993 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
10 vcrel 23925 . . . . . . 7  |-  Rel  CVecOLD
11 1st2nd 6620 . . . . . . 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 23981 . . . . . . 7  |-  G  =  ( 1st `  ( 1st `  U ) )
15 phop.4 . . . . . . . 8  |-  S  =  ( .sOLD `  U )
1615smfval 23983 . . . . . . 7  |-  S  =  ( 2nd `  ( 1st `  U ) )
1714, 16opeq12i 4064 . . . . . 6  |-  <. G ,  S >.  =  <. ( 1st `  ( 1st `  U
) ) ,  ( 2nd `  ( 1st `  U ) ) >.
1812, 17syl6eqr 2493 . . . . 5  |-  ( ( 1st `  U )  e.  CVecOLD  ->  ( 1st `  U )  = 
<. G ,  S >. )
197, 9, 183syl 20 . . . 4  |-  ( U  e.  CPreHil OLD  ->  ( 1st `  U )  =  <. G ,  S >. )
2019opeq1d 4065 . . 3  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  N >.  =  <. <. G ,  S >. ,  N >. )
216, 20syl5eqr 2489 . 2  |-  ( U  e.  CPreHil OLD  ->  <. ( 1st `  U ) ,  ( 2nd `  U
) >.  =  <. <. G ,  S >. ,  N >. )
223, 21eqtrd 2475 1  |-  ( U  e.  CPreHil OLD  ->  U  = 
<. <. G ,  S >. ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3883   Rel wrel 4845   ` cfv 5418   1stc1st 6575   2ndc2nd 6576   CVecOLDcvc 23923   NrmCVeccnv 23962   +vcpv 23963   .sOLDcns 23965   normCVcnmcv 23968   CPreHil OLDccphlo 24212
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-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-1st 6577  df-2nd 6578  df-vc 23924  df-nv 23970  df-va 23973  df-ba 23974  df-sm 23975  df-0v 23976  df-nmcv 23978  df-ph 24213
This theorem is referenced by:  phpar  24224
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