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Theorem limsupval 12197
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 2909 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 5140 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,)  +oo ) )  =  ( F " (
k [,)  +oo ) ) )
32ineq1d 3486 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,)  +oo ) )  i^i  RR* )  =  ( ( F " (
k [,)  +oo ) )  i^i  RR* ) )
43supeq1d 7388 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
54mpteq2dv 4239 . . . . . 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 2439 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 5039 . . . 4  |-  ( x  =  F  ->  ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7388 . . 3  |-  ( x  =  F  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
10 df-limsup 12194 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 10668 . . . . 5  |-  <  Or  RR*
12 cnvso 5353 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 200 . . . 4  |-  `'  <  Or 
RR*
1413supex 7403 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5747 . 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 set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264    e. cmpt 4209    Or wor 4445   `'ccnv 4819   ran crn 4821   "cima 4823   ` cfv 5396  (class class class)co 6022   supcsup 7382   RRcr 8924    +oocpnf 9052   RR*cxr 9054    < clt 9055   [,)cico 10852   limsupclsp 12193
This theorem is referenced by:  limsuple  12201  limsupval2  12203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-pre-lttri 8999  ax-pre-lttrn 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-limsup 12194
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