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Theorem limsuplt 13384
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 13383 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
32notbid 292 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
) )
4 rexnal 2902 . . 3  |-  ( E. j  e.  RR  -.  A  <_  ( G `  j )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
)
53, 4syl6bbr 263 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  E. j  e.  RR  -.  A  <_ 
( G `  j
) ) )
6 simp2 995 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
7 reex 9572 . . . . . . 7  |-  RR  e.  _V
87ssex 4581 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
983ad2ant1 1015 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
10 xrex 11218 . . . . . 6  |-  RR*  e.  _V
1110a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
12 fex2 6728 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
136, 9, 11, 12syl3anc 1226 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F  e.  _V )
14 limsupcl 13378 . . . 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 996 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 xrltnle 9642 . . 3  |-  ( ( ( limsup `  F )  e.  RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
1815, 16, 17syl2anc 659 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
191limsupgf 13380 . . . . . 6  |-  G : RR
--> RR*
2019ffvelrni 6006 . . . . 5  |-  ( j  e.  RR  ->  ( G `  j )  e.  RR* )
2120adantl 464 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  ( G `  j )  e.  RR* )
22 simpl3 999 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  A  e.  RR* )
23 xrltnle 9642 . . . 4  |-  ( ( ( G `  j
)  e.  RR*  /\  A  e.  RR* )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2421, 22, 23syl2anc 659 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2524rexbidva 2962 . 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 285 1  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  E. j  e.  RR  ( G `  j )  <  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   _Vcvv 3106    i^i cin 3460    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618   [,)cico 11534   limsupclsp 13375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-limsup 13376
This theorem is referenced by:  limsupgre  13386
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