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Theorem caurcvgr 12146
Description: A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that  F is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.)
Hypotheses
Ref Expression
caurcvgr.1  |-  ( ph  ->  A  C_  RR )
caurcvgr.2  |-  ( ph  ->  F : A --> RR )
caurcvgr.3  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
caurcvgr.4  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) )
Assertion
Ref Expression
caurcvgr  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Distinct variable groups:    j, k, x, A    j, F, k, x    ph, j, k, x

Proof of Theorem caurcvgr
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 caurcvgr.1 . . . . 5  |-  ( ph  ->  A  C_  RR )
2 caurcvgr.2 . . . . 5  |-  ( ph  ->  F : A --> RR )
3 caurcvgr.3 . . . . 5  |-  ( ph  ->  sup ( A ,  RR* ,  <  )  = 
+oo )
4 caurcvgr.4 . . . . 5  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( F `  j ) ) )  <  x ) )
5 1rp 10358 . . . . . 6  |-  1  e.  RR+
65a1i 10 . . . . 5  |-  ( ph  ->  1  e.  RR+ )
71, 2, 3, 4, 6caucvgrlem 12145 . . . 4  |-  ( ph  ->  E. j  e.  A  ( ( limsup `  F
)  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  1 ) ) ) )
8 simpl 443 . . . . 5  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
98rexlimivw 2663 . . . 4  |-  ( E. j  e.  A  ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
107, 9syl 15 . . 3  |-  ( ph  ->  ( limsup `  F )  e.  RR )
1110recnd 8861 . 2  |-  ( ph  ->  ( limsup `  F )  e.  CC )
121adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
132adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : A
--> RR )
143adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
154adantr 451 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  A. x  e.  RR+  E. j  e.  A  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( F `  j )
) )  <  x
) )
16 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
17 3re 9817 . . . . . . . . 9  |-  3  e.  RR
18 3pos 9830 . . . . . . . . 9  |-  0  <  3
1917, 18elrpii 10357 . . . . . . . 8  |-  3  e.  RR+
20 rpdivcl 10376 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  3  e.  RR+ )  ->  (
y  /  3 )  e.  RR+ )
2116, 19, 20sylancl 643 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  /  3 )  e.  RR+ )
2212, 13, 14, 15, 21caucvgrlem 12145 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  A  ( ( limsup `
 F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) ) )
23 simpr 447 . . . . . . 7  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )  ->  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
2423reximi 2650 . . . . . 6  |-  ( E. j  e.  A  ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )  ->  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
2522, 24syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  A  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
26 ssrexv 3238 . . . . 5  |-  ( A 
C_  RR  ->  ( E. j  e.  A  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  (
y  /  3 ) ) )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) ) )
2712, 25, 26sylc 56 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) ) )
28 rpcn 10362 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e.  CC )
2928adantl 452 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  CC )
30 3cn 9818 . . . . . . . . 9  |-  3  e.  CC
3130a1i 10 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  e.  CC )
32 3ne0 9831 . . . . . . . . 9  |-  3  =/=  0
3332a1i 10 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  =/=  0 )
3429, 31, 33divcan2d 9538 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( 3  x.  ( y  / 
3 ) )  =  y )
3534breq2d 4035 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( abs `  ( ( F `
 k )  -  ( limsup `  F )
) )  <  (
3  x.  ( y  /  3 ) )  <-> 
( abs `  (
( F `  k
)  -  ( limsup `  F ) ) )  <  y ) )
3635imbi2d 307 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
( 3  x.  (
y  /  3 ) ) )  <->  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) ) )
3736rexralbidv 2587 . . . 4  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  ( y  / 
3 ) ) )  <->  E. j  e.  RR  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
y ) ) )
3827, 37mpbid 201 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) )
3938ralrimiva 2626 . 2  |-  ( ph  ->  A. y  e.  RR+  E. j  e.  RR  A. k  e.  A  (
j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `  F
) ) )  < 
y ) )
40 ax-resscn 8794 . . . 4  |-  RR  C_  CC
41 fss 5397 . . . 4  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
422, 40, 41sylancl 643 . . 3  |-  ( ph  ->  F : A --> CC )
43 eqidd 2284 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  =  ( F `  k ) )
4442, 1, 43rlim 11969 . 2  |-  ( ph  ->  ( F  ~~> r  (
limsup `  F )  <->  ( ( limsup `
 F )  e.  CC  /\  A. y  e.  RR+  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) ) ) )
4511, 39, 44mpbir2and 888 1  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   3c3 9796   RR+crp 10354   abscabs 11719   limsupclsp 11944    ~~> r crli 11959
This theorem is referenced by:  caucvgrlem2  12147  caurcvg  12149
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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  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-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-rlim 11963
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