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Theorem limsupval 11878
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
StepHypRef Expression
1 elex 2748 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 4960 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,)  +oo ) )  =  ( F " (
k [,)  +oo ) ) )
32ineq1d 3311 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,)  +oo ) )  i^i  RR* )  =  ( ( F " (
k [,)  +oo ) )  i^i  RR* ) )
43supeq1d 7132 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
54mpteq2dv 4047 . . . . . 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 2306 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 4859 . . . 4  |-  ( x  =  F  ->  ran  (  k  e.  RR  |->  sup ( ( ( x
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7132 . . 3  |-  ( x  =  F  ->  sup ( ran  (  k  e.  RR  |->  sup ( ( ( x " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
10 df-limsup 11875 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  (  k  e.  RR  |->  sup (
( ( x "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 10407 . . . . 5  |-  <  Or  RR*
12 cnvso 5166 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 201 . . . 4  |-  `'  <  Or 
RR*
1413supex 7147 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5501 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
161, 15syl 17 1  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   _Vcvv 2740    i^i cin 3093    e. cmpt 4017    Or wor 4250   `'ccnv 4625   ran crn 4627   "cima 4629   ` cfv 4638  (class class class)co 5757   supcsup 7126   RRcr 8669    +oocpnf 8797   RR*cxr 8799    < clt 8800   [,)cico 10589   limsupclsp 11874
This theorem is referenced by:  limsuple  11882  limsupval2  11884
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-cnex 8726  ax-resscn 8727  ax-pre-lttri 8744  ax-pre-lttrn 8745
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-sup 7127  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-limsup 11875
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