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Theorem limsupval 12957
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 2986 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 imaeq1 5169 . . . . . . . . 9  |-  ( x  =  F  ->  (
x " ( k [,) +oo ) )  =  ( F "
( k [,) +oo ) ) )
32ineq1d 3556 . . . . . . . 8  |-  ( x  =  F  ->  (
( x " (
k [,) +oo )
)  i^i  RR* )  =  ( ( F "
( k [,) +oo ) )  i^i  RR* ) )
43supeq1d 7701 . . . . . . 7  |-  ( x  =  F  ->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " (
k [,) +oo )
)  i^i  RR* ) , 
RR* ,  <  ) )
54mpteq2dv 4384 . . . . . 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 2493 . . . . 5  |-  ( x  =  F  ->  (
k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  G )
87rneqd 5072 . . . 4  |-  ( x  =  F  ->  ran  ( k  e.  RR  |->  sup ( ( ( x
" ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ran  G
)
98supeq1d 7701 . . 3  |-  ( x  =  F  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( x " ( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) , 
RR* ,  `'  <  )  =  sup ( ran 
G ,  RR* ,  `'  <  ) )
10 df-limsup 12954 . . 3  |-  limsup  =  ( x  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( x "
( k [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
11 xrltso 11123 . . . . 5  |-  <  Or  RR*
12 cnvso 5381 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
1311, 12mpbi 208 . . . 4  |-  `'  <  Or 
RR*
1413supex 7718 . . 3  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  _V
159, 10, 14fvmpt 5779 . 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 1369    e. wcel 1756   _Vcvv 2977    i^i cin 3332    e. cmpt 4355    Or wor 4645   `'ccnv 4844   ran crn 4846   "cima 4848   ` cfv 5423  (class class class)co 6096   supcsup 7695   RRcr 9286   +oocpnf 9420   RR*cxr 9422    < clt 9423   [,)cico 11307   limsupclsp 12953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-limsup 12954
This theorem is referenced by:  limsuple  12961  limsupval2  12963
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