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Theorem limsupval 13308
 Description: The superior limit of an infinite sequence of extended real numbers, which is the infimum (indicated by ) of the set of suprema of all upper infinite subsequences of . Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 5-Sep-2014.)
Hypothesis
Ref Expression
limsupval.1
Assertion
Ref Expression
limsupval
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem limsupval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 imaeq1 5342 . . . . . . . . 9
32ineq1d 3695 . . . . . . . 8
43supeq1d 7923 . . . . . . 7
54mpteq2dv 4544 . . . . . 6
6 limsupval.1 . . . . . 6
75, 6syl6eqr 2516 . . . . 5
87rneqd 5240 . . . 4
98supeq1d 7923 . . 3
10 df-limsup 13305 . . 3
11 xrltso 11372 . . . . 5
12 cnvso 5552 . . . . 5
1311, 12mpbi 208 . . . 4
1413supex 7940 . . 3
159, 10, 14fvmpt 5956 . 2
161, 15syl 16 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1395   wcel 1819  cvv 3109   cin 3470   cmpt 4515   wor 4808  ccnv 5007   crn 5009  cima 5011  cfv 5594  (class class class)co 6296  csup 7918  cr 9508   cpnf 9642  cxr 9644   clt 9645  cico 11556  clsp 13304 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-limsup 13305 This theorem is referenced by:  limsuple  13312  limsupval2  13314
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