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Theorem caurcvg 12425
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 10461 . . . . . 6  |-  ( ZZ>= `  M )  C_  ZZ
31, 2eqsstri 3338 . . . . 5  |-  Z  C_  ZZ
4 zssre 10245 . . . . 5  |-  ZZ  C_  RR
53, 4sstri 3317 . . . 4  |-  Z  C_  RR
65a1i 11 . . 3  |-  ( ph  ->  Z  C_  RR )
7 caurcvg.3 . . 3  |-  ( ph  ->  F : Z --> RR )
8 1rp 10572 . . . . . 6  |-  1  e.  RR+
9 ne0i 3594 . . . . . 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 3677 . . . . 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 10449 . . . . . . . . 9  |-  ( m  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1514, 1eleq2s 2496 . . . . . . . 8  |-  ( m  e.  Z  ->  M  e.  ZZ )
161uzsup 11199 . . . . . . . 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 2784 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  sup ( Z ,  RR* ,  <  )  = 
+oo )
2019rexlimivw 2786 . . . 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 3304 . . . . . . . . . . . 12  |-  ( m  e.  Z  ->  m  e.  ZZ )
233sseli 3304 . . . . . . . . . . . 12  |-  ( k  e.  Z  ->  k  e.  ZZ )
24 eluz 10455 . . . . . . . . . . . 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 2746 . . . . . 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 2771 . . . . 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 2741 . . . 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 12422 . 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 2784 . . . . 5  |-  ( E. m  e.  Z  A. k  e.  ( ZZ>= `  m ) ( abs `  ( ( F `  k )  -  ( F `  m )
) )  <  x  ->  M  e.  ZZ )
3736rexlimivw 2786 . . . 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 9003 . . . 4  |-  RR  C_  CC
40 fss 5558 . . . 4  |-  ( ( F : Z --> RR  /\  RR  C_  CC )  ->  F : Z --> CC )
417, 39, 40sylancl 644 . . 3  |-  ( ph  ->  F : Z --> CC )
421, 38, 41rlimclim 12295 . 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 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   1c1 8947    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   abscabs 11994   limsupclsp 12219    ~~> cli 12233    ~~> r crli 12234
This theorem is referenced by:  caurcvg2  12426  mbflimlem  19512
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fl 11157  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238
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