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Theorem limsupval2 11905
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
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 11899 . . 3  |-  ( F  e.  V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
41, 3syl 17 . 2  |-  ( ph  ->  ( limsup `  F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
5 imassrn 4999 . . . . 5  |-  ( G
" A )  C_  ran  G
62limsupgf 11900 . . . . . . 7  |-  G : RR
--> RR*
7 frn 5319 . . . . . . 7  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
86, 7ax-mp 10 . . . . . 6  |-  ran  G  C_ 
RR*
9 infmxrlb 10604 . . . . . . 7  |-  ( ( ran  G  C_  RR*  /\  x  e.  ran  G )  ->  sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
109ralrimiva 2599 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
118, 10mp1i 13 . . . . 5  |-  ( ph  ->  A. x  e.  ran  G sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
12 ssralv 3198 . . . . 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 3149 . . . . 5  |-  ( G
" A )  C_  RR*
15 infmxrcl 10587 . . . . . 6  |-  ( ran 
G  C_  RR*  ->  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR* )
168, 15ax-mp 10 . . . . 5  |-  sup ( ran  G ,  RR* ,  `'  <  )  e.  RR*
17 infmxrgelb 10605 . . . . 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 656 . . . 4  |-  ( sup ( ran  G ,  RR* ,  `'  <  )  <_  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <->  A. x  e.  ( G " A ) sup ( ran  G ,  RR* ,  `'  <  )  <_  x )
1913, 18sylibr 205 . . 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 8830 . . . . . . . . 9  |-  RR  C_  RR*
2321, 22syl6ss 3152 . . . . . . . 8  |-  ( ph  ->  A  C_  RR* )
24 supxrunb1 10590 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2523, 24syl 17 . . . . . . 7  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
2620, 25mpbird 225 . . . . . 6  |-  ( ph  ->  A. n  e.  RR  E. x  e.  A  n  <_  x )
27 infmxrcl 10587 . . . . . . . . . . 11  |-  ( ( G " A ) 
C_  RR*  ->  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR* )
2814, 27mp1i 13 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  e.  RR* )
2921sselda 3141 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  RR )
3029ad2ant2r 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  RR )
316ffvelrni 5584 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  ( G `  x )  e.  RR* )
3230, 31syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  RR* )
336ffvelrni 5584 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  ( G `  n )  e.  RR* )
3433ad2antlr 710 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  e.  RR* )
35 ffn 5313 . . . . . . . . . . . . 13  |-  ( G : RR --> RR*  ->  G  Fn  RR )
366, 35mp1i 13 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  G  Fn  RR )
3721ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  A  C_  RR )
38 simprl 735 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  x  e.  A
)
39 fnfvima 5676 . . . . . . . . . . . 12  |-  ( ( G  Fn  RR  /\  A  C_  RR  /\  x  e.  A )  ->  ( G `  x )  e.  ( G " A
) )
4036, 37, 38, 39syl3anc 1187 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  e.  ( G " A ) )
41 infmxrlb 10604 . . . . . . . . . . 11  |-  ( ( ( G " A
)  C_  RR*  /\  ( G `  x )  e.  ( G " A
) )  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  x )
)
4214, 40, 41sylancr 647 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  x ) )
43 simplr 734 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  e.  RR )
44 simprr 736 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  n  <_  x
)
45 limsupgord 11897 . . . . . . . . . . . 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 1187 . . . . . . . . . . 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 11901 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  ( G `  x )  =  sup ( ( ( F " ( x [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
4830, 47syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  =  sup ( ( ( F
" ( x [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
492limsupgval 11901 . . . . . . . . . . . 12  |-  ( n  e.  RR  ->  ( G `  n )  =  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5049ad2antlr 710 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  n )  =  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
5146, 48, 503brtr4d 4013 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  ( G `  x )  <_  ( G `  n )
)
5228, 32, 34, 42, 51xrletrd 10446 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  RR )  /\  (
x  e.  A  /\  n  <_  x ) )  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  ( G `  n ) )
5352expr 601 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  RR )  /\  x  e.  A )  ->  (
n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5453rexlimdva 2640 . . . . . . 7  |-  ( (
ph  /\  n  e.  RR )  ->  ( E. x  e.  A  n  <_  x  ->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5554ralimdva 2594 . . . . . 6  |-  ( ph  ->  ( A. n  e.  RR  E. x  e.  A  n  <_  x  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5626, 55mpd 16 . . . . 5  |-  ( ph  ->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
576, 35ax-mp 10 . . . . . 6  |-  G  Fn  RR
58 breq2 3987 . . . . . . 7  |-  ( x  =  ( G `  n )  ->  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
) )
5958ralrn 5588 . . . . . 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 10 . . . . 5  |-  ( A. x  e.  ran  G sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  x  <->  A. n  e.  RR  sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  ( G `  n )
)
6156, 60sylibr 205 . . . 4  |-  ( ph  ->  A. x  e.  ran  G sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  x )
6214, 27ax-mp 10 . . . . 5  |-  sup (
( G " A
) ,  RR* ,  `'  <  )  e.  RR*
63 infmxrgelb 10605 . . . . 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 656 . . . 4  |-  ( sup ( ( G " A ) ,  RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G sup (
( G " A
) ,  RR* ,  `'  <  )  <_  x )
6561, 64sylibr 205 . . 3  |-  ( ph  ->  sup ( ( G
" A ) , 
RR* ,  `'  <  )  <_  sup ( ran  G ,  RR* ,  `'  <  ) )
66 xrletri3 10439 . . . 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 656 . . 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 648 . 2  |-  ( ph  ->  sup ( ran  G ,  RR* ,  `'  <  )  =  sup ( ( G " A ) ,  RR* ,  `'  <  ) )
694, 68eqtrd 2288 1  |-  ( ph  ->  ( limsup `  F )  =  sup ( ( G
" A ) , 
RR* ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517    i^i cin 3112    C_ wss 3113   class class class wbr 3983    e. cmpt 4037   `'ccnv 4646   ran crn 4648   "cima 4650    Fn wfn 4654   -->wf 4655   ` cfv 4659  (class class class)co 5778   supcsup 7147   RRcr 8690    +oocpnf 8818   RR*cxr 8820    < clt 8821    <_ cle 8822   [,)cico 10610   limsupclsp 11895
This theorem is referenced by:  mbflimsup  18969
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-sup 7148  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-ico 10614  df-limsup 11896
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