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Theorem phop 22324
 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
phop.4
phop.6 CV
Assertion
Ref Expression
phop

Proof of Theorem phop
StepHypRef Expression
1 phrel 22321 . . 3
2 1st2nd 6396 . . 3
31, 2mpan 653 . 2
4 phop.6 . . . . 5 CV
54nmcvfval 22091 . . . 4
65opeq2i 3990 . . 3
7 phnv 22320 . . . . 5
8 eqid 2438 . . . . . 6
98nvvc 22099 . . . . 5
10 vcrel 22031 . . . . . . 7
11 1st2nd 6396 . . . . . . 7
1210, 11mpan 653 . . . . . 6
13 phop.2 . . . . . . . 8
1413vafval 22087 . . . . . . 7
15 phop.4 . . . . . . . 8
1615smfval 22089 . . . . . . 7
1714, 16opeq12i 3991 . . . . . 6
1812, 17syl6eqr 2488 . . . . 5
197, 9, 183syl 19 . . . 4
2019opeq1d 3992 . . 3
216, 20syl5eqr 2484 . 2
223, 21eqtrd 2470 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  cop 3819   wrel 4886  cfv 5457  c1st 6350  c2nd 6351  cvc 22029  cnv 22068  cpv 22069  cns 22071  CVcnmcv 22074  ccphlo 22318 This theorem is referenced by:  phpar  22330 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-1st 6352  df-2nd 6353  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084  df-ph 22319
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