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Theorem limsupval 12268
 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 2964 . 2
2 imaeq1 5198 . . . . . . . . 9
32ineq1d 3541 . . . . . . . 8
43supeq1d 7451 . . . . . . 7
54mpteq2dv 4296 . . . . . 6
6 limsupval.1 . . . . . 6
75, 6syl6eqr 2486 . . . . 5
87rneqd 5097 . . . 4
98supeq1d 7451 . . 3
10 df-limsup 12265 . . 3
11 xrltso 10734 . . . . 5
12 cnvso 5411 . . . . 5
1311, 12mpbi 200 . . . 4
1413supex 7468 . . 3
159, 10, 14fvmpt 5806 . 2
161, 15syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2956   cin 3319   cmpt 4266   wor 4502  ccnv 4877   crn 4879  cima 4881  cfv 5454  (class class class)co 6081  csup 7445  cr 8989   cpnf 9117  cxr 9119   clt 9120  cico 10918  clsp 12264 This theorem is referenced by:  limsuple  12272  limsupval2  12274 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-limsup 12265
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