MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limsupcl Unicode version

Theorem limsupcl 11947
Description: Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by Mario Carneiro, 7-May-2016.)
Assertion
Ref Expression
limsupcl  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )

Proof of Theorem limsupcl
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( F  e.  V  ->  F  e.  _V )
2 df-limsup 11945 . . . 4  |-  limsup  =  ( f  e.  _V  |->  sup ( ran  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  ) )
3 eqid 2283 . . . . . . 7  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4 inss2 3390 . . . . . . . 8  |-  ( ( f " ( k [,)  +oo ) )  i^i  RR* )  C_  RR*
5 supxrcl 10633 . . . . . . . 8  |-  ( ( ( f " (
k [,)  +oo ) )  i^i  RR* )  C_  RR*  ->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
64, 5mp1i 11 . . . . . . 7  |-  ( k  e.  RR  ->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
73, 6fmpti 5683 . . . . . 6  |-  ( k  e.  RR  |->  sup (
( ( f "
( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
8 frn 5395 . . . . . 6  |-  ( ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*  ->  ran  ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR* )
97, 8ax-mp 8 . . . . 5  |-  ran  (
k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR*
10 infmxrcl 10635 . . . . 5  |-  ( ran  ( k  e.  RR  |->  sup ( ( ( f
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  C_  RR*  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  e.  RR* )
119, 10mp1i 11 . . . 4  |-  ( f  e.  _V  ->  sup ( ran  ( k  e.  RR  |->  sup ( ( ( f " ( k [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) ,  RR* ,  `'  <  )  e.  RR* )
122, 11fmpti 5683 . . 3  |-  limsup : _V --> RR*
1312ffvelrni 5664 . 2  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
141, 13syl 15 1  |-  ( F  e.  V  ->  ( limsup `
 F )  e. 
RR* )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152    e. cmpt 4077   `'ccnv 4688   ran crn 4690   "cima 4692   -->wf 5251   ` 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:  limsuplt  11953  limsupbnd1  11956  caucvgrlem  12145
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-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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-reu 2550  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  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-le 8873  df-sub 9039  df-neg 9040  df-limsup 11945
  Copyright terms: Public domain W3C validator