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Theorem caurcvgr 12394
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 10548 . . . . . 6  |-  1  e.  RR+
65a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR+ )
71, 2, 3, 4, 6caucvgrlem 12393 . . . 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 444 . . . . 5  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
98rexlimivw 2769 . . . 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 16 . . 3  |-  ( ph  ->  ( limsup `  F )  e.  RR )
1110recnd 9047 . 2  |-  ( ph  ->  ( limsup `  F )  e.  CC )
121adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
132adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : A
--> RR )
143adantr 452 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( A ,  RR* ,  <  )  =  +oo )
154adantr 452 . . . . . . 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 448 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
17 3re 10003 . . . . . . . . 9  |-  3  e.  RR
18 3pos 10016 . . . . . . . . 9  |-  0  <  3
1917, 18elrpii 10547 . . . . . . . 8  |-  3  e.  RR+
20 rpdivcl 10566 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  3  e.  RR+ )  ->  (
y  /  3 )  e.  RR+ )
2116, 19, 20sylancl 644 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  /  3 )  e.  RR+ )
2212, 13, 14, 15, 21caucvgrlem 12393 . . . . . 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 448 . . . . . . 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 2756 . . . . . 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 16 . . . . 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 3351 . . . . 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 58 . . . 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 10552 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e.  CC )
2928adantl 453 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  CC )
30 3cn 10004 . . . . . . . . 9  |-  3  e.  CC
3130a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  e.  CC )
32 3ne0 10017 . . . . . . . . 9  |-  3  =/=  0
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  =/=  0 )
3429, 31, 33divcan2d 9724 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( 3  x.  ( y  / 
3 ) )  =  y )
3534breq2d 4165 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( abs `  ( ( F `
 k )  -  ( limsup `  F )
) )  <  (
3  x.  ( y  /  3 ) )  <-> 
( abs `  (
( F `  k
)  -  ( limsup `  F ) ) )  <  y ) )
3635imbi2d 308 . . . . 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 2693 . . . 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 202 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) )
3938ralrimiva 2732 . 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 8980 . . . 4  |-  RR  C_  CC
41 fss 5539 . . . 4  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
422, 40, 41sylancl 644 . . 3  |-  ( ph  ->  F : A --> CC )
43 eqidd 2388 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  =  ( F `  k ) )
4442, 1, 43rlim 12216 . 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 889 1  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   E.wrex 2650    C_ wss 3263   class class class wbr 4153   -->wf 5390   ` cfv 5394  (class class class)co 6020   supcsup 7380   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    +oocpnf 9050   RR*cxr 9052    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   3c3 9982   RR+crp 10544   abscabs 11966   limsupclsp 12191    ~~> r crli 12206
This theorem is referenced by:  caucvgrlem2  12395  caurcvg  12397
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-ico 10854  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-rlim 12210
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