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Theorem nvop 24244
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 24159 . . 3  |-  Rel  NrmCVec
2 1st2nd 6733 . . 3  |-  ( ( Rel  NrmCVec  /\  U  e.  NrmCVec )  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
31, 2mpan 670 . 2  |-  ( U  e.  NrmCVec  ->  U  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >. )
4 nvop.6 . . . . 5  |-  N  =  ( normCV `  U )
54nmcvfval 24164 . . . 4  |-  N  =  ( 2nd `  U
)
65opeq2i 4174 . . 3  |-  <. ( 1st `  U ) ,  N >.  =  <. ( 1st `  U ) ,  ( 2nd `  U
) >.
7 eqid 2454 . . . . 5  |-  ( 1st `  U )  =  ( 1st `  U )
8 nvop.2 . . . . 5  |-  G  =  ( +v `  U
)
9 nvop.4 . . . . 5  |-  S  =  ( .sOLD `  U )
107, 8, 9nvvop 24166 . . . 4  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  =  <. G ,  S >. )
1110opeq1d 4176 . . 3  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  N >.  = 
<. <. G ,  S >. ,  N >. )
126, 11syl5eqr 2509 . 2  |-  ( U  e.  NrmCVec  ->  <. ( 1st `  U
) ,  ( 2nd `  U ) >.  =  <. <. G ,  S >. ,  N >. )
133, 12eqtrd 2495 1  |-  ( U  e.  NrmCVec  ->  U  =  <. <. G ,  S >. ,  N >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   <.cop 3994   Rel wrel 4956   ` cfv 5529   1stc1st 6688   2ndc2nd 6689   NrmCVeccnv 24141   +vcpv 24142   .sOLDcns 24144   normCVcnmcv 24147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-oprab 6207  df-1st 6690  df-2nd 6691  df-vc 24103  df-nv 24149  df-va 24152  df-sm 24154  df-nmcv 24157
This theorem is referenced by:  sspval  24300  isph  24401  hilhhi  24745
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