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Theorem limsuplt 13061
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 13060 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
32notbid 294 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( -.  A  <_  ( limsup `  F )  <->  -.  A. j  e.  RR  A  <_  ( G `  j )
) )
4 rexnal 2842 . . 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 989 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
7 reex 9476 . . . . . . 7  |-  RR  e.  _V
87ssex 4536 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
983ad2ant1 1009 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
10 xrex 11091 . . . . . 6  |-  RR*  e.  _V
1110a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
12 fex2 6634 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
136, 9, 11, 12syl3anc 1219 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F  e.  _V )
14 limsupcl 13055 . . . 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 990 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 xrltnle 9546 . . 3  |-  ( ( ( limsup `  F )  e.  RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
1815, 16, 17syl2anc 661 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  (
( limsup `  F )  <  A  <->  -.  A  <_  (
limsup `  F ) ) )
191limsupgf 13057 . . . . . 6  |-  G : RR
--> RR*
2019ffvelrni 5943 . . . . 5  |-  ( j  e.  RR  ->  ( G `  j )  e.  RR* )
2120adantl 466 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  ( G `  j )  e.  RR* )
22 simpl3 993 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  A  e.  RR* )
23 xrltnle 9546 . . . 4  |-  ( ( ( G `  j
)  e.  RR*  /\  A  e.  RR* )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2421, 22, 23syl2anc 661 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e.  RR* )  /\  j  e.  RR )  ->  (
( G `  j
)  <  A  <->  -.  A  <_  ( G `  j
) ) )
2524rexbidva 2845 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   _Vcvv 3070    i^i cin 3427    C_ wss 3428   class class class wbr 4392    |-> cmpt 4450   "cima 4943   -->wf 5514   ` cfv 5518  (class class class)co 6192   supcsup 7793   RRcr 9384   +oocpnf 9518   RR*cxr 9520    < clt 9521    <_ cle 9522   [,)cico 11405   limsupclsp 13052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-sup 7794  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-limsup 13053
This theorem is referenced by:  limsupgre  13063
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