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Theorem limsuplt 12275
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 12274 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
32notbid 287 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
) )
4 rexnal 2718 . . 3  |-  ( E. j  e.  RR  -.  A  <_  ( G `  j )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
)
53, 4syl6bbr 256 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  E. j  e.  RR  -.  A  <_ 
( G `  j
) ) )
6 simp2 959 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
7 reex 9083 . . . . . . 7  |-  RR  e.  _V
87ssex 4349 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
983ad2ant1 979 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
10 xrex 10611 . . . . . 6  |-  RR*  e.  _V
1110a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
12 fex2 5605 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
136, 9, 11, 12syl3anc 1185 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F  e.  _V )
14 limsupcl 12269 . . . 4  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
1513, 14syl 16 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( limsup `
 F )  e. 
RR* )
16 simp3 960 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 xrltnle 9146 . . 3  |-  ( ( ( limsup `  F )  e.  RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
1815, 16, 17syl2anc 644 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
191limsupgf 12271 . . . . . 6  |-  G : RR
--> RR*
2019ffvelrni 5871 . . . . 5  |-  ( j  e.  RR  ->  ( G `  j )  e.  RR* )
2120adantl 454 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  ( G `  j )  e.  RR* )
22 simpl3 963 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  A  e.  RR* )
23 xrltnle 9146 . . . 4  |-  ( ( ( G `  j
)  e.  RR*  /\  A  e.  RR* )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2421, 22, 23syl2anc 644 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2524rexbidva 2724 . 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 278 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   supcsup 7447   RRcr 8991    +oocpnf 9119   RR*cxr 9121    < clt 9122    <_ cle 9123   [,)cico 10920   limsupclsp 12266
This theorem is referenced by:  limsupgre  12277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-limsup 12267
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