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Theorem caurcvg 12433
Description: A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that  F is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 8-May-2016.)
Hypotheses
Ref Expression
caurcvg.1  |-  Z  =  ( ZZ>= `  M )
caurcvg.3  |-  ( ph  ->  F : Z --> RR )
caurcvg.4  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
Assertion
Ref Expression
caurcvg  |-  ( ph  ->  F  ~~>  ( limsup `  F
) )
Distinct variable groups:    k, m, x, F    m, M, x    ph, k, m, x    k, Z, m, x
Allowed substitution hint:    M( k)

Proof of Theorem caurcvg
StepHypRef Expression
1 caurcvg.1 . . . . . 6  |-  Z  =  ( ZZ>= `  M )
2 uzssz 10469 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3346 . . . . 5  |-  Z  C_  ZZ
4 zssre 10253 . . . . 5  |-  ZZ  C_  RR
53, 4sstri 3325 . . . 4  |-  Z  C_  RR
65a1i 11 . . 3  |-  ( ph  ->  Z  C_  RR )
7 caurcvg.3 . . 3  |-  ( ph  ->  F : Z --> RR )
8 1rp 10580 . . . . . 6  |-  1  e.  RR+
9 ne0i 3602 . . . . . 6  |-  ( 1  e.  RR+  ->  RR+  =/=  (/) )
108, 9ax-mp 8 . . . . 5  |-  RR+  =/=  (/)
11 caurcvg.4 . . . . 5  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
12 r19.2z 3685 . . . . 5  |-  ( (
RR+  =/=  (/)  /\  A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
1310, 11, 12sylancr 645 . . . 4  |-  ( ph  ->  E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
)
14 eluzel2 10457 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1514, 1eleq2s 2504 . . . . . . . 8  |-  ( m  e.  Z  ->  M  e.  ZZ )
161uzsup 11207 . . . . . . . 8  |-  ( M  e.  ZZ  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1715, 16syl 16 . . . . . . 7  |-  ( m  e.  Z  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
1817a1d 23 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  sup ( Z ,  RR* ,  <  )  =  +oo ) )
1918rexlimiv 2792 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
2019rexlimivw 2794 . . . 4  |-  ( E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  sup ( Z ,  RR* ,  <  )  =  +oo )
2113, 20syl 16 . . 3  |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
223sseli 3312 . . . . . . . . . . . 12  |-  ( m  e.  Z  ->  m  e.  ZZ )
233sseli 3312 . . . . . . . . . . . 12  |-  ( k  e.  Z  ->  k  e.  ZZ )
24 eluz 10463 . . . . . . . . . . . 12  |-  ( ( m  e.  ZZ  /\  k  e.  ZZ )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2522, 23, 24syl2an 464 . . . . . . . . . . 11  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( k  e.  (
ZZ>= `  m )  <->  m  <_  k ) )
2625biimprd 215 . . . . . . . . . 10  |-  ( ( m  e.  Z  /\  k  e.  Z )  ->  ( m  <_  k  ->  k  e.  ( ZZ>= `  m ) ) )
2726expimpd 587 . . . . . . . . 9  |-  ( m  e.  Z  ->  (
( k  e.  Z  /\  m  <_  k )  ->  k  e.  (
ZZ>= `  m ) ) )
2827imim1d 71 . . . . . . . 8  |-  ( m  e.  Z  ->  (
( k  e.  (
ZZ>= `  m )  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  ( ( k  e.  Z  /\  m  <_  k )  ->  ( abs `  ( ( F `
 k )  -  ( F `  m ) ) )  <  x
) ) )
2928exp4a 590 . . . . . . 7  |-  ( m  e.  Z  ->  (
( k  e.  (
ZZ>= `  m )  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x )  ->  ( k  e.  Z  ->  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) ) ) )
3029ralimdv2 2754 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  A. k  e.  Z  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) ) )
3130reximia 2779 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  E. m  e.  Z  A. k  e.  Z  ( m  <_  k  -> 
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x ) )
3231ralimi 2749 . . . 4  |-  ( A. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  Z  ( m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x
) )
3311, 32syl 16 . . 3  |-  ( ph  ->  A. x  e.  RR+  E. m  e.  Z  A. k  e.  Z  (
m  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x ) )
346, 7, 21, 33caurcvgr 12430 . 2  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
3515a1d 23 . . . . . 6  |-  ( m  e.  Z  ->  ( A. k  e.  ( ZZ>=
`  m ) ( abs `  ( ( F `  k )  -  ( F `  m ) ) )  <  x  ->  M  e.  ZZ ) )
3635rexlimiv 2792 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  M  e.  ZZ )
3736rexlimivw 2794 . . . 4  |-  ( E. x  e.  RR+  E. m  e.  Z  A. k  e.  ( ZZ>= `  m )
( abs `  (
( F `  k
)  -  ( F `
 m ) ) )  <  x  ->  M  e.  ZZ )
3813, 37syl 16 . . 3  |-  ( ph  ->  M  e.  ZZ )
39 ax-resscn 9011 . . . 4  |-  RR  C_  CC
40 fss 5566 . . . 4  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
417, 39, 40sylancl 644 . . 3  |-  ( ph  ->  F : Z --> CC )
421, 38, 41rlimclim 12303 . 2  |-  ( ph  ->  ( F  ~~> r  (
limsup `  F )  <->  F  ~~>  ( limsup `  F ) ) )
4334, 42mpbid 202 1  |-  ( ph  ->  F  ~~>  ( limsup `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675    C_ wss 3288   (/)c0 3596   class class class wbr 4180   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   CCcc 8952   RRcr 8953   1c1 8955    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085    - cmin 9255   ZZcz 10246   ZZ>=cuz 10452   RR+crp 10576   abscabs 12002   limsupclsp 12227    ~~> cli 12241    ~~> r crli 12242
This theorem is referenced by:  caurcvg2  12434  mbflimlem  19520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-ico 10886  df-fl 11165  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-limsup 12228  df-clim 12245  df-rlim 12246
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