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Theorem limsupval2 11954
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 11948 . . 3  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
41, 3syl 15 . 2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
5 imassrn 5025 . . . . 5  |-  ( G
" A )  C_  ran  G
62limsupgf 11949 . . . . . . 7  |-  G : RR
--> RR*
7 frn 5395 . . . . . . 7  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
86, 7ax-mp 8 . . . . . 6  |-  ran  G  C_ 
RR*
9 infmxrlb 10652 . . . . . . 7  |-  ( ( ran  G  C_  RR*  /\  x  e.  ran  G )  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
109ralrimiva 2626 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
118, 10mp1i 11 . . . . 5  |-  ( ph  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
12 ssralv 3237 . . . . 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 59 . . . 4  |-  ( ph  ->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
145, 8sstri 3188 . . . . 5  |-  ( G
" A )  C_  RR*
15 infmxrcl 10635 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )
168, 15ax-mp 8 . . . . 5  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*
17 infmxrgelb 10653 . . . . 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 653 . . . 4  |-  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
1913, 18sylibr 203 . . 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 8876 . . . . . . . . 9  |-  RR  C_  RR*
2321, 22syl6ss 3191 . . . . . . . 8  |-  ( ph  ->  A  C_  RR* )
24 supxrunb1 10638 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2523, 24syl 15 . . . . . . 7  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2620, 25mpbird 223 . . . . . 6  |-  ( ph  ->  A. n  e.  RR  E. x  e.  A  n  <_  x )
27 infmxrcl 10635 . . . . . . . . . . 11  |-  ( ( G " A ) 
C_  RR*  ->  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR* )
2814, 27mp1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  e.  RR* )
2921sselda 3180 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3029ad2ant2r 727 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  RR )
316ffvelrni 5664 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  ( G `  x )  e.  RR* )
3230, 31syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  RR* )
336ffvelrni 5664 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  ( G `  n )  e.  RR* )
3433ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  e.  RR* )
35 ffn 5389 . . . . . . . . . . . . 13  |-  ( G : RR --> RR*  ->  G  Fn  RR )
366, 35mp1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  G  Fn  RR )
3721ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  A  C_  RR )
38 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  A
)
39 fnfvima 5756 . . . . . . . . . . . 12  |-  ( ( G  Fn  RR  /\  A  C_  RR  /\  x  e.  A )  ->  ( G `  x )  e.  ( G " A
) )
4036, 37, 38, 39syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  ( G " A ) )
41 infmxrlb 10652 . . . . . . . . . . 11  |-  ( ( ( G " A
)  C_  RR*  /\  ( G `  x )  e.  ( G " A
) )  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  x )
)
4214, 40, 41sylancr 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  x ) )
43 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  e.  RR )
44 simprr 733 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  <_  x
)
45 limsupgord 11946 . . . . . . . . . . . 12  |-  ( ( 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 1182 . . . . . . . . . . 11  |-  ( ( ( 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 11950 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  ( G `  x )  =  sup ( ( ( F " ( x [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4830, 47syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  =  sup ( ( ( F
" ( x [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
492limsupgval 11950 . . . . . . . . . . . 12  |-  ( n  e.  RR  ->  ( G `  n )  =  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5049ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  =  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5146, 48, 503brtr4d 4053 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  <_  ( G `  n )
)
5228, 32, 34, 42, 51xrletrd 10493 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  n ) )
5352expr 598 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  RR )  /\  x  e.  A )  ->  (
n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5453rexlimdva 2667 . . . . . . 7  |-  ( (
ph  /\  n  e.  RR )  ->  ( E. x  e.  A  n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5554ralimdva 2621 . . . . . 6  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5626, 55mpd 14 . . . . 5  |-  ( ph  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
576, 35ax-mp 8 . . . . . 6  |-  G  Fn  RR
58 breq2 4027 . . . . . . 7  |-  ( x  =  ( G `  n )  ->  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5958ralrn 5668 . . . . . 6  |-  ( G  Fn  RR  ->  ( A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n ) ) )
6057, 59ax-mp 8 . . . . 5  |-  ( A. x  e.  ran  G sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
6156, 60sylibr 203 . . . 4  |-  ( ph  ->  A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x )
6214, 27ax-mp 8 . . . . 5  |-  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR*
63 infmxrgelb 10653 . . . . 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 ) )
648, 62, 63mp2an 653 . . . 4  |-  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G sup (
( G " A
) ,  RR* ,  `'  <  )  <_  x )
6561, 64sylibr 203 . . 3  |-  ( ph  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) )
66 xrletri3 10486 . . . 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* ,  `'  <  ) ) ) )
6716, 62, 66mp2an 653 . . 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* ,  `'  <  ) ) )
6819, 65, 67sylanbrc 645 . 2  |-  ( ph  ->  sup ( ran  G ,  RR* ,  `'  <  )  =  sup ( ( G " A ) ,  RR* ,  `'  <  ) )
694, 68eqtrd 2315 1  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658   limsupclsp 11944
This theorem is referenced by:  mbflimsup  19021
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-iun 3907  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-1st 6122  df-2nd 6123  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-ico 10662  df-limsup 11945
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