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Theorem nvop 25980
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  =  ( .sOLD `  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 25895 . . 3  |-  Rel  NrmCVec
2 1st2nd 6829 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 668 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 25900 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 4162 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 eqid 2402 . . . . 5  |-  ( 1st `  U )  =  ( 1st `  U )
8 nvop.2 . . . . 5  |-  G  =  ( +v `  U
)
9 nvop.4 . . . . 5  |-  S  =  ( .sOLD `  U )
107, 8, 9nvvop 25902 . . . 4  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
1110opeq1d 4164 . . 3  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
126, 11syl5eqr 2457 . 2  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  =  <. <. G ,  S >. ,  N >. )
133, 12eqtrd 2443 1  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   <.cop 3977   Rel wrel 4827   ` cfv 5568   1stc1st 6781   2ndc2nd 6782   NrmCVeccnv 25877   +vcpv 25878   .sOLDcns 25880   normCVcnmcv 25883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-oprab 6281  df-1st 6783  df-2nd 6784  df-vc 25839  df-nv 25885  df-va 25888  df-sm 25890  df-nmcv 25893
This theorem is referenced by:  sspval  26036  isph  26137  hilhhi  26481
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