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Theorem nvvop 22090
 Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvvop.1
nvvop.2
nvvop.4
Assertion
Ref Expression
nvvop

Proof of Theorem nvvop
StepHypRef Expression
1 vcrel 22028 . . 3
2 nvss 22074 . . . . 5
3 nvvop.1 . . . . . . . 8
4 eqid 2438 . . . . . . . 8 CV CV
53, 4nvop2 22089 . . . . . . 7 CV
65eleq1d 2504 . . . . . 6 CV
76ibi 234 . . . . 5 CV
82, 7sseldi 3348 . . . 4 CV
9 opelxp1 4913 . . . 4 CV
108, 9syl 16 . . 3
11 1st2nd 6395 . . 3
121, 10, 11sylancr 646 . 2
13 nvvop.2 . . . . 5
1413vafval 22084 . . . 4
153fveq2i 5733 . . . 4
1614, 15eqtr4i 2461 . . 3
17 nvvop.4 . . . . 5
1817smfval 22086 . . . 4
193fveq2i 5733 . . . 4
2018, 19eqtr4i 2461 . . 3
2116, 20opeq12i 3991 . 2
2212, 21syl6eqr 2488 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  cvv 2958  cop 3819   cxp 4878   wrel 4885  cfv 5456  c1st 6349  c2nd 6350  cvc 22026  cnv 22065  cpv 22066  cns 22068  CVcnmcv 22071 This theorem is referenced by:  nvi  22095  nvvc  22096  nvop  22168 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 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-rab 2716  df-v 2960  df-sbc 3164  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-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-oprab 6087  df-1st 6351  df-2nd 6352  df-vc 22027  df-nv 22073  df-va 22076  df-sm 22078  df-nmcv 22081
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