MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limsupval2 Unicode version

Theorem limsupval2 11970
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 11964 . . 3  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
41, 3syl 15 . 2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
5 imassrn 5041 . . . . 5  |-  ( G
" A )  C_  ran  G
62limsupgf 11965 . . . . . . 7  |-  G : RR
--> RR*
7 frn 5411 . . . . . . 7  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
86, 7ax-mp 8 . . . . . 6  |-  ran  G  C_ 
RR*
9 infmxrlb 10668 . . . . . . 7  |-  ( ( ran  G  C_  RR*  /\  x  e.  ran  G )  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
109ralrimiva 2639 . . . . . 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 3250 . . . . 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 3201 . . . . 5  |-  ( G
" A )  C_  RR*
15 infmxrcl 10651 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )
168, 15ax-mp 8 . . . . 5  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*
17 infmxrgelb 10669 . . . . 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 8892 . . . . . . . . 9  |-  RR  C_  RR*
2321, 22syl6ss 3204 . . . . . . . 8  |-  ( ph  ->  A  C_  RR* )
24 supxrunb1 10654 . . . . . . . 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 10651 . . . . . . . . . . 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 3193 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3029ad2ant2r 727 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  RR )
316ffvelrni 5680 . . . . . . . . . . 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 5680 . . . . . . . . . . 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 5405 . . . . . . . . . . . . 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 5772 . . . . . . . . . . . 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 10668 . . . . . . . . . . 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 11962 . . . . . . . . . . . 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 11966 . . . . . . . . . . . 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 11966 . . . . . . . . . . . 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 4069 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  <_  ( G `  n )
)
5228, 32, 34, 42, 51xrletrd 10509 . . . . . . . . 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 2680 . . . . . . 7  |-  ( (
ph  /\  n  e.  RR )  ->  ( E. x  e.  A  n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5554ralimdva 2634 . . . . . 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 4043 . . . . . . 7  |-  ( x  =  ( G `  n )  ->  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5958ralrn 5684 . . . . . 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 10669 . . . . 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 10502 . . . 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 2328 1  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   [,)cico 10674   limsupclsp 11960
This theorem is referenced by:  mbflimsup  19037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ico 10678  df-limsup 11961
  Copyright terms: Public domain W3C validator