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Theorem limsupval2 13282
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 13276 . . 3  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
41, 3syl 16 . 2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
5 imassrn 5354 . . . . 5  |-  ( G
" A )  C_  ran  G
62limsupgf 13277 . . . . . . 7  |-  G : RR
--> RR*
7 frn 5743 . . . . . . 7  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
86, 7ax-mp 5 . . . . . 6  |-  ran  G  C_ 
RR*
9 infmxrlb 11537 . . . . . . 7  |-  ( ( ran  G  C_  RR*  /\  x  e.  ran  G )  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
109ralrimiva 2881 . . . . . 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 3569 . . . . 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 63 . . . 4  |-  ( ph  ->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
145, 8sstri 3518 . . . . 5  |-  ( G
" A )  C_  RR*
15 infmxrcl 11520 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )
168, 15ax-mp 5 . . . . 5  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*
17 infmxrgelb 11538 . . . . 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 672 . . . 4  |-  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
1913, 18sylibr 212 . . 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 9649 . . . . . . . . 9  |-  RR  C_  RR*
2321, 22syl6ss 3521 . . . . . . . 8  |-  ( ph  ->  A  C_  RR* )
24 supxrunb1 11523 . . . . . . . 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 232 . . . . . 6  |-  ( ph  ->  A. n  e.  RR  E. x  e.  A  n  <_  x )
27 infmxrcl 11520 . . . . . . . . . 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 3509 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3029ad2ant2r 746 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  RR )
316ffvelrni 6031 . . . . . . . . . 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 6031 . . . . . . . . . 10  |-  ( n  e.  RR  ->  ( G `  n )  e.  RR* )
3433ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  e.  RR* )
35 ffn 5737 . . . . . . . . . . . 12  |-  ( G : RR --> RR*  ->  G  Fn  RR )
366, 35mp1i 12 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  G  Fn  RR )
3721ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  A  C_  RR )
38 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  A
)
39 fnfvima 6149 . . . . . . . . . . 11  |-  ( ( G  Fn  RR  /\  A  C_  RR  /\  x  e.  A )  ->  ( G `  x )  e.  ( G " A
) )
4036, 37, 38, 39syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  ( G " A ) )
41 infmxrlb 11537 . . . . . . . . . 10  |-  ( ( ( G " A
)  C_  RR*  /\  ( G `  x )  e.  ( G " A
) )  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  x )
)
4214, 40, 41sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  x ) )
43 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  e.  RR )
44 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  <_  x
)
45 limsupgord 13274 . . . . . . . . . . 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 1228 . . . . . . . . . 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 13278 . . . . . . . . . . 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 13278 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  ( G `  n )  =  sup ( ( ( F " ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5049ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  =  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5146, 48, 503brtr4d 4483 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  <_  ( G `  n )
)
5228, 32, 34, 42, 51xrletrd 11377 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  n ) )
5352rexlimdvaa 2960 . . . . . . 7  |-  ( (
ph  /\  n  e.  RR )  ->  ( E. x  e.  A  n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5453ralimdva 2875 . . . . . 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 5 . . . . . 6  |-  G  Fn  RR
57 breq2 4457 . . . . . . 7  |-  ( x  =  ( G `  n )  ->  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5857ralrn 6035 . . . . . 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 5 . . . . 5  |-  ( A. x  e.  ran  G sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
6055, 59sylibr 212 . . . 4  |-  ( ph  ->  A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x )
6114, 27ax-mp 5 . . . . 5  |-  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR*
62 infmxrgelb 11538 . . . . 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 672 . . . 4  |-  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G sup (
( G " A
) ,  RR* ,  `'  <  )  <_  x )
6460, 63sylibr 212 . . 3  |-  ( ph  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) )
65 xrletri3 11370 . . . 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 672 . . 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 664 . 2  |-  ( ph  ->  sup ( ran  G ,  RR* ,  `'  <  )  =  sup ( ( G " A ) ,  RR* ,  `'  <  ) )
684, 67eqtrd 2508 1  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    i^i cin 3480    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   ran crn 5006   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   +oocpnf 9637   RR*cxr 9639    < clt 9640    <_ cle 9641   [,)cico 11543   limsupclsp 13272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-ico 11547  df-limsup 13273
This theorem is referenced by:  mbflimsup  21939
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