MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caurcvgr Structured version   Unicode version

Theorem caurcvgr 13253
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 11096 . . . . . 6  |-  1  e.  RR+
65a1i 11 . . . . 5  |-  ( ph  ->  1  e.  RR+ )
71, 2, 3, 4, 6caucvgrlem 13252 . . . 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 457 . . . . 5  |-  ( ( ( limsup `  F )  e.  RR  /\  A. k  e.  A  ( j  <_  k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  ( 3  x.  1 ) ) )  ->  ( limsup `  F )  e.  RR )
98rexlimivw 2933 . . . 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 9513 . 2  |-  ( ph  ->  ( limsup `  F )  e.  CC )
121adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  RR+ )  ->  A  C_  RR )
132adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  F : A
--> RR )
143adantr 465 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  sup ( A ,  RR* ,  <  )  = +oo )
154adantr 465 . . . . . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
17 3re 10496 . . . . . . . . 9  |-  3  e.  RR
18 3pos 10516 . . . . . . . . 9  |-  0  <  3
1917, 18elrpii 11095 . . . . . . . 8  |-  3  e.  RR+
20 rpdivcl 11114 . . . . . . . 8  |-  ( ( y  e.  RR+  /\  3  e.  RR+ )  ->  (
y  /  3 )  e.  RR+ )
2116, 19, 20sylancl 662 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( y  /  3 )  e.  RR+ )
2212, 13, 14, 15, 21caucvgrlem 13252 . . . . . 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 461 . . . . . . 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 2919 . . . . . 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 3515 . . . . 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 60 . . . 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 11100 . . . . . . . . 9  |-  ( y  e.  RR+  ->  y  e.  CC )
2928adantl 466 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  CC )
30 3cn 10497 . . . . . . . . 9  |-  3  e.  CC
3130a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  e.  CC )
32 3ne0 10517 . . . . . . . . 9  |-  3  =/=  0
3332a1i 11 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR+ )  ->  3  =/=  0 )
3429, 31, 33divcan2d 10210 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( 3  x.  ( y  / 
3 ) )  =  y )
3534breq2d 4402 . . . . . 6  |-  ( (
ph  /\  y  e.  RR+ )  ->  ( ( abs `  ( ( F `
 k )  -  ( limsup `  F )
) )  <  (
3  x.  ( y  /  3 ) )  <-> 
( abs `  (
( F `  k
)  -  ( limsup `  F ) ) )  <  y ) )
3635imbi2d 316 . . . . 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 2863 . . . 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 210 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. j  e.  RR  A. k  e.  A  ( j  <_ 
k  ->  ( abs `  ( ( F `  k )  -  ( limsup `
 F ) ) )  <  y ) )
3938ralrimiva 2822 . 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 9440 . . . 4  |-  RR  C_  CC
41 fss 5665 . . . 4  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
422, 40, 41sylancl 662 . . 3  |-  ( ph  ->  F : A --> CC )
43 eqidd 2452 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  ( F `  k )  =  ( F `  k ) )
4442, 1, 43rlim 13075 . 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 913 1  |-  ( ph  ->  F  ~~> r  ( limsup `  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796    C_ wss 3426   class class class wbr 4390   -->wf 5512   ` cfv 5516  (class class class)co 6190   supcsup 7791   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384    x. cmul 9388   +oocpnf 9516   RR*cxr 9518    < clt 9519    <_ cle 9520    - cmin 9696    / cdiv 10094   3c3 10473   RR+crp 11092   abscabs 12825   limsupclsp 13050    ~~> r crli 13065
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 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-2nd 6678  df-recs 6932  df-rdg 6966  df-er 7201  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ico 11407  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-rlim 13069
This theorem is referenced by:  caucvgrlem2  13254  caurcvg  13256
  Copyright terms: Public domain W3C validator