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Theorem limsuplt 11969
Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
Hypothesis
Ref Expression
limsupval.1  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Assertion
Ref Expression
limsuplt  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  E. j  e.  RR  ( G `  j )  <  A ) )
Distinct variable groups:    A, j    B, j    j, G    j,
k, F
Allowed substitution hints:    A( k)    B( k)    G( k)

Proof of Theorem limsuplt
StepHypRef Expression
1 limsupval.1 . . . . 5  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
21limsuple 11968 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
32notbid 285 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
) )
4 rexnal 2567 . . 3  |-  ( E. j  e.  RR  -.  A  <_  ( G `  j )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
)
53, 4syl6bbr 254 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  E. j  e.  RR  -.  A  <_ 
( G `  j
) ) )
6 simp2 956 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
7 reex 8844 . . . . . . 7  |-  RR  e.  _V
87ssex 4174 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
983ad2ant1 976 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
10 xrex 10367 . . . . . 6  |-  RR*  e.  _V
1110a1i 10 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
12 fex2 5417 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
136, 9, 11, 12syl3anc 1182 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F  e.  _V )
14 limsupcl 11963 . . . 4  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
1513, 14syl 15 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( limsup `
 F )  e. 
RR* )
16 simp3 957 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 xrltnle 8907 . . 3  |-  ( ( ( limsup `  F )  e.  RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
1815, 16, 17syl2anc 642 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
191limsupgf 11965 . . . . . 6  |-  G : RR
--> RR*
2019ffvelrni 5680 . . . . 5  |-  ( j  e.  RR  ->  ( G `  j )  e.  RR* )
2120adantl 452 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  ( G `  j )  e.  RR* )
22 simpl3 960 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  A  e.  RR* )
23 xrltnle 8907 . . . 4  |-  ( ( ( G `  j
)  e.  RR*  /\  A  e.  RR* )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2421, 22, 23syl2anc 642 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2524rexbidva 2573 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( E. j  e.  RR  ( G `  j )  <  A  <->  E. j  e.  RR  -.  A  <_ 
( G `  j
) ) )
265, 18, 253bitr4d 276 1  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  E. j  e.  RR  ( G `  j )  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   "cima 4708   -->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:  limsupgre  11971
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-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-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-limsup 11961
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