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Theorem nvss 22077
 Description: Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
nvss

Proof of Theorem nvss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2498 . . . . . . 7
21biimpar 473 . . . . . 6
323ad2antr1 1123 . . . . 5 GId
43exlimivv 1646 . . . 4 GId
5 vex 2961 . . . 4
64, 5jctir 526 . . 3 GId
76ssopab2i 4485 . 2 GId
8 df-nv 22076 . . 3 GId
9 dfoprab2 6124 . . 3 GId GId
108, 9eqtri 2458 . 2 GId
11 df-xp 4887 . 2
127, 10, 113sstr4i 3389 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  wral 2707  cvv 2958   wss 3322  cop 3819   class class class wbr 4215  copab 4268   cxp 4879   crn 4882  wf 5453  cfv 5457  (class class class)co 6084  coprab 6085  cc 8993  cr 8994  cc0 8995   caddc 8998   cmul 9000   cle 9126  cabs 12044  GIdcgi 21780  cvc 22029  cnv 22068 This theorem is referenced by:  nvvcop  22078  nvrel  22086  nvvop  22093  nvex  22095 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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  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-opab 4270  df-xp 4887  df-oprab 6088  df-nv 22076
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