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Theorem limsuplt 11953
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 11952 . . . 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 2554 . . 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 8828 . . . . . . 7  |-  RR  e.  _V
87ssex 4158 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
983ad2ant1 976 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
10 xrex 10351 . . . . . 6  |-  RR*  e.  _V
1110a1i 10 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
12 fex2 5401 . . . . 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 11947 . . . 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 8891 . . 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 11949 . . . . . 6  |-  G : RR
--> RR*
2019ffvelrni 5664 . . . . 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 8891 . . . 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 2560 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   "cima 4692   -->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:  limsupgre  11955
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-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-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-limsup 11945
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