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Theorem nvi 22085
 Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvi.1
nvi.2
nvi.4
nvi.5
nvi.6 CV
Assertion
Ref Expression
nvi
Distinct variable groups:   ,,   ,,   ,   ,,   ,,
Allowed substitution hints:   ()   (,)

Proof of Theorem nvi
StepHypRef Expression
1 eqid 2435 . . . . . 6
2 nvi.6 . . . . . 6 CV
31, 2nvop2 22079 . . . . 5
4 nvi.2 . . . . . . 7
5 nvi.4 . . . . . . 7
61, 4, 5nvvop 22080 . . . . . 6
76opeq1d 3982 . . . . 5
83, 7eqtrd 2467 . . . 4
9 id 20 . . . 4
108, 9eqeltrrd 2510 . . 3
11 nvi.1 . . . . 5
1211, 4bafval 22075 . . . 4
13 eqid 2435 . . . 4 GId GId
1412, 13isnv 22083 . . 3 GId
1510, 14sylib 189 . 2 GId
16 nvi.5 . . . . . . . 8
174, 160vfval 22077 . . . . . . 7 GId
1817eqeq2d 2446 . . . . . 6 GId
1918imbi2d 308 . . . . 5 GId
20193anbi1d 1258 . . . 4 GId
2120ralbidv 2717 . . 3 GId
22213anbi3d 1260 . 2 GId
2315, 22mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725  wral 2697  cop 3809   class class class wbr 4204  wf 5442  cfv 5446  (class class class)co 6073  c1st 6339  cc 8980  cr 8981  cc0 8982   caddc 8985   cmul 8987   cle 9113  cabs 12031  GIdcgi 21767  cvc 22016  cnv 22055  cpv 22056  cba 22057  cns 22058  cn0v 22059  CVcnmcv 22061 This theorem is referenced by:  nvvc  22086  nvf  22139  nvs  22143  nvz  22150  nvtri  22151 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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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