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Theorem limsupval2 12229
Description: The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 8-May-2016.)
Hypotheses
Ref Expression
limsupval.1  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
limsupval2.1  |-  ( ph  ->  F  e.  V )
limsupval2.2  |-  ( ph  ->  A  C_  RR )
limsupval2.3  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
Assertion
Ref Expression
limsupval2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Distinct variable group:    k, F
Allowed substitution hints:    ph( k)    A( k)    G( k)    V( k)

Proof of Theorem limsupval2
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupval2.1 . . 3  |-  ( ph  ->  F  e.  V )
2 limsupval.1 . . . 4  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
32limsupval 12223 . . 3  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
41, 3syl 16 . 2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
5 imassrn 5175 . . . . 5  |-  ( G
" A )  C_  ran  G
62limsupgf 12224 . . . . . . 7  |-  G : RR
--> RR*
7 frn 5556 . . . . . . 7  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
86, 7ax-mp 8 . . . . . 6  |-  ran  G  C_ 
RR*
9 infmxrlb 10868 . . . . . . 7  |-  ( ( ran  G  C_  RR*  /\  x  e.  ran  G )  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
109ralrimiva 2749 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
118, 10mp1i 12 . . . . 5  |-  ( ph  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
12 ssralv 3367 . . . . 5  |-  ( ( G " A ) 
C_  ran  G  ->  ( A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x  ->  A. x  e.  ( G " A
) sup ( ran 
G ,  RR* ,  `'  <  )  <_  x )
)
135, 11, 12mpsyl 61 . . . 4  |-  ( ph  ->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
145, 8sstri 3317 . . . . 5  |-  ( G
" A )  C_  RR*
15 infmxrcl 10851 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )
168, 15ax-mp 8 . . . . 5  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*
17 infmxrgelb 10869 . . . . 5  |-  ( ( ( G " A
)  C_  RR*  /\  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )  ->  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup (
( G " A
) ,  RR* ,  `'  <  )  <->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x ) )
1814, 16, 17mp2an 654 . . . 4  |-  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
1913, 18sylibr 204 . . 3  |-  ( ph  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
20 limsupval2.3 . . . . . . 7  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
21 limsupval2.2 . . . . . . . . 9  |-  ( ph  ->  A  C_  RR )
22 ressxr 9085 . . . . . . . . 9  |-  RR  C_  RR*
2321, 22syl6ss 3320 . . . . . . . 8  |-  ( ph  ->  A  C_  RR* )
24 supxrunb1 10854 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2523, 24syl 16 . . . . . . 7  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2620, 25mpbird 224 . . . . . 6  |-  ( ph  ->  A. n  e.  RR  E. x  e.  A  n  <_  x )
27 infmxrcl 10851 . . . . . . . . . 10  |-  ( ( G " A ) 
C_  RR*  ->  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR* )
2814, 27mp1i 12 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  e.  RR* )
2921sselda 3308 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3029ad2ant2r 728 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  RR )
316ffvelrni 5828 . . . . . . . . . 10  |-  ( x  e.  RR  ->  ( G `  x )  e.  RR* )
3230, 31syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  RR* )
336ffvelrni 5828 . . . . . . . . . 10  |-  ( n  e.  RR  ->  ( G `  n )  e.  RR* )
3433ad2antlr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  e.  RR* )
35 ffn 5550 . . . . . . . . . . . 12  |-  ( G : RR --> RR*  ->  G  Fn  RR )
366, 35mp1i 12 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  G  Fn  RR )
3721ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  A  C_  RR )
38 simprl 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  A
)
39 fnfvima 5935 . . . . . . . . . . 11  |-  ( ( G  Fn  RR  /\  A  C_  RR  /\  x  e.  A )  ->  ( G `  x )  e.  ( G " A
) )
4036, 37, 38, 39syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  ( G " A ) )
41 infmxrlb 10868 . . . . . . . . . 10  |-  ( ( ( G " A
)  C_  RR*  /\  ( G `  x )  e.  ( G " A
) )  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  x )
)
4214, 40, 41sylancr 645 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  x ) )
43 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  e.  RR )
44 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  <_  x
)
45 limsupgord 12221 . . . . . . . . . . 11  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  n  <_  x )  ->  sup ( ( ( F
" ( x [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4643, 30, 44, 45syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( ( F " ( x [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
472limsupgval 12225 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  ( G `  x )  =  sup ( ( ( F " ( x [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4830, 47syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  =  sup ( ( ( F
" ( x [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
492limsupgval 12225 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  ( G `  n )  =  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5049ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  =  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5146, 48, 503brtr4d 4202 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  <_  ( G `  n )
)
5228, 32, 34, 42, 51xrletrd 10708 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  n ) )
5352rexlimdvaa 2791 . . . . . . 7  |-  ( (
ph  /\  n  e.  RR )  ->  ( E. x  e.  A  n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5453ralimdva 2744 . . . . . 6  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5526, 54mpd 15 . . . . 5  |-  ( ph  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
566, 35ax-mp 8 . . . . . 6  |-  G  Fn  RR
57 breq2 4176 . . . . . . 7  |-  ( x  =  ( G `  n )  ->  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5857ralrn 5832 . . . . . 6  |-  ( G  Fn  RR  ->  ( A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n ) ) )
5956, 58ax-mp 8 . . . . 5  |-  ( A. x  e.  ran  G sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
6055, 59sylibr 204 . . . 4  |-  ( ph  ->  A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x )
6114, 27ax-mp 8 . . . . 5  |-  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR*
62 infmxrgelb 10869 . . . . 5  |-  ( ( ran  G  C_  RR*  /\  sup ( ( G " A ) ,  RR* ,  `'  <  )  e.  RR* )  ->  ( sup (
( G " A
) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x ) )
638, 61, 62mp2an 654 . . . 4  |-  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G sup (
( G " A
) ,  RR* ,  `'  <  )  <_  x )
6460, 63sylibr 204 . . 3  |-  ( ph  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) )
65 xrletri3 10701 . . . 4  |-  ( ( sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*  /\  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR* )  ->  ( sup ( ran 
G ,  RR* ,  `'  <  )  =  sup (
( G " A
) ,  RR* ,  `'  <  )  <->  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup (
( G " A
) ,  RR* ,  `'  <  )  /\  sup (
( G " A
) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) ) ) )
6616, 61, 65mp2an 654 . . 3  |-  ( sup ( ran  G ,  RR* ,  `'  <  )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <-> 
( sup ( ran 
G ,  RR* ,  `'  <  )  <_  sup (
( G " A
) ,  RR* ,  `'  <  )  /\  sup (
( G " A
) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) ) )
6719, 64, 66sylanbrc 646 . 2  |-  ( ph  ->  sup ( ran  G ,  RR* ,  `'  <  )  =  sup ( ( G " A ) ,  RR* ,  `'  <  ) )
684, 67eqtrd 2436 1  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667    i^i cin 3279    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   [,)cico 10874   limsupclsp 12219
This theorem is referenced by:  mbflimsup  19511
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-ico 10878  df-limsup 12220
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