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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
nvop.4
nvop.6 CV
Assertion
Ref Expression
nvop

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 22074 . . 3
2 1st2nd 6386 . . 3
31, 2mpan 652 . 2
4 nvop.6 . . . . 5 CV
54nmcvfval 22079 . . . 4
65opeq2i 3981 . . 3
7 eqid 2436 . . . . 5
8 nvop.2 . . . . 5
9 nvop.4 . . . . 5
107, 8, 9nvvop 22081 . . . 4
1110opeq1d 3983 . . 3
126, 11syl5eqr 2482 . 2
133, 12eqtrd 2468 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cop 3810   wrel 4876  cfv 5447  c1st 6340  c2nd 6341  cnv 22056  cpv 22057  cns 22059  CVcnmcv 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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