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Theorem limsupval 11948
Description: The superior limit of an infinite sequence  F of extended real numbers, which is the infimum (indicated by  `'  <) of the set of suprema of all upper infinite subsequences of  F. 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  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Assertion
Ref Expression
limsupval  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Distinct variable group:    k, F
Allowed substitution hints:    G( k)    V( k)

Proof of Theorem limsupval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 5007 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,)  +oo ) )  =  ( F " (
k [,)  +oo ) ) )
32ineq1d 3369 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,)  +oo ) )  i^i  RR* )  =  ( ( F " (
k [,)  +oo ) )  i^i  RR* ) )
43supeq1d 7199 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
54mpteq2dv 4107 . . . . . 6  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( k  e.  RR  |->  sup (
( ( F "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) )
6 limsupval.1 . . . . . 6  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
75, 6syl6eqr 2333 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 4906 . . . 4  |-  ( x  =  F  ->  ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7199 . . 3  |-  ( x  =  F  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
10 df-limsup 11945 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 10475 . . . . 5  |-  <  Or  RR*
12 cnvso 5214 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 199 . . . 4  |-  `'  <  Or 
RR*
1413supex 7214 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5602 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
161, 15syl 15 1  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077    Or wor 4313   `'ccnv 4688   ran crn 4690   "cima 4692   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867   [,)cico 10658   limsupclsp 11944
This theorem is referenced by:  limsuple  11952  limsupval2  11954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-limsup 11945
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