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Theorem nmosetre 22266
 Description: The set in the supremum of the operator norm definition df-nmoo 22247 is a set of reals. (Contributed by NM, 13-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmosetre.2
nmosetre.4 CV
Assertion
Ref Expression
nmosetre
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem nmosetre
StepHypRef Expression
1 ffvelrn 5869 . . . . . . . . 9
2 nmosetre.2 . . . . . . . . . 10
3 nmosetre.4 . . . . . . . . . 10 CV
42, 3nvcl 22149 . . . . . . . . 9
51, 4sylan2 462 . . . . . . . 8
65anassrs 631 . . . . . . 7
7 eleq1 2497 . . . . . . 7
86, 7syl5ibr 214 . . . . . 6
98impcom 421 . . . . 5
109adantrl 698 . . . 4
1110exp31 589 . . 3
1211rexlimdv 2830 . 2
1312abssdv 3418 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cab 2423  wrex 2707   wss 3321   class class class wbr 4213  wf 5451  cfv 5455  cr 8990  c1 8992   cle 9122  cnv 22064  cba 22066  CVcnmcv 22070 This theorem is referenced by:  nmoxr  22268  nmooge0  22269  nmorepnf  22270  nmoolb  22273  nmoubi  22274  nmlno0lem  22295  nmopsetretHIL  23368 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-1st 6350  df-2nd 6351  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-nmcv 22080
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