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Theorem limsupval 13308
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 3118 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 5342 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,) +oo ) )  =  ( F "
( k [,) +oo ) ) )
32ineq1d 3695 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,) +oo )
)  i^i  RR* )  =  ( ( F "
( k [,) +oo ) )  i^i  RR* ) )
43supeq1d 7923 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,) +oo )
)  i^i  RR* ) , 
RR* ,  <  ) )
54mpteq2dv 4544 . . . . . 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 2516 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 5240 . . . 4  |-  ( x  =  F  ->  ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7923 . . 3  |-  ( x  =  F  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  `'  <  )  =  sup ( ran 
G ,  RR* ,  `'  <  ) )
10 df-limsup 13305 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 11372 . . . . 5  |-  <  Or  RR*
12 cnvso 5552 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 208 . . . 4  |-  `'  <  Or 
RR*
1413supex 7940 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5956 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
161, 15syl 16 1  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    |-> cmpt 4515    Or wor 4808   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645   [,)cico 11556   limsupclsp 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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