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

Theorem limsupgval 11997
Description: Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
Hypothesis
Ref Expression
limsupval.1  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Assertion
Ref Expression
limsupgval  |-  ( M  e.  RR  ->  ( G `  M )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Distinct variable groups:    k, F    k, M
Allowed substitution hint:    G( k)

Proof of Theorem limsupgval
StepHypRef Expression
1 oveq1 5907 . . . . 5  |-  ( k  =  M  ->  (
k [,)  +oo )  =  ( M [,)  +oo ) )
21imaeq2d 5049 . . . 4  |-  ( k  =  M  ->  ( F " ( k [,) 
+oo ) )  =  ( F " ( M [,)  +oo ) ) )
32ineq1d 3403 . . 3  |-  ( k  =  M  ->  (
( F " (
k [,)  +oo ) )  i^i  RR* )  =  ( ( F " ( M [,)  +oo ) )  i^i  RR* ) )
43supeq1d 7244 . 2  |-  ( k  =  M  ->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5 limsupval.1 . 2  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
6 xrltso 10522 . . 3  |-  <  Or  RR*
76supex 7259 . 2  |-  sup (
( ( F "
( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  _V
84, 5, 7fvmpt 5640 1  |-  ( M  e.  RR  ->  ( G `  M )  =  sup ( ( ( F " ( M [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    i^i cin 3185    e. cmpt 4114   "cima 4729   ` cfv 5292  (class class class)co 5900   supcsup 7238   RRcr 8781    +oocpnf 8909   RR*cxr 8911    < clt 8912   [,)cico 10705
This theorem is referenced by:  limsupgle  11998  limsupval2  12001  limsupgre  12002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-pre-lttri 8856  ax-pre-lttrn 8857
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917
  Copyright terms: Public domain W3C validator