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Theorem climsqz2 13119
Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
Hypotheses
Ref Expression
climadd.1  |-  Z  =  ( ZZ>= `  M )
climadd.2  |-  ( ph  ->  M  e.  ZZ )
climadd.4  |-  ( ph  ->  F  ~~>  A )
climsqz.5  |-  ( ph  ->  G  e.  W )
climsqz.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
climsqz.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
climsqz2.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k
) )
climsqz2.9  |-  ( (
ph  /\  k  e.  Z )  ->  A  <_  ( G `  k
) )
Assertion
Ref Expression
climsqz2  |-  ( ph  ->  G  ~~>  A )
Distinct variable groups:    k, F    ph, k    A, k    k, G   
k, M    k, Z
Allowed substitution hint:    W( k)

Proof of Theorem climsqz2
Dummy variables  x  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climadd.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
2 climadd.2 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
32adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  ZZ )
4 simpr 461 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5 eqidd 2444 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
6 climadd.4 . . . . . 6  |-  ( ph  ->  F  ~~>  A )
76adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  F  ~~>  A )
81, 3, 4, 5, 7climi2 12989 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  x )
91uztrn2 10878 . . . . . . . 8  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
10 climsqz.7 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
11 climsqz.6 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
121, 2, 6, 11climrecl 13061 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
1312adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
14 climsqz2.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k
) )
1510, 11, 13, 14lesub1dd 9955 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  <_  ( ( F `
 k )  -  A ) )
16 climsqz2.9 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  <_  ( G `  k
) )
1713, 10, 16abssubge0d 12918 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  =  ( ( G `  k )  -  A
) )
1813, 10, 11, 16, 14letrd 9528 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
) )
1913, 11, 18abssubge0d 12918 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( ( F `  k )  -  A
) )
2015, 17, 193brtr4d 4322 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2120adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2210adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
2312ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  A  e.  RR )
2422, 23resubcld 9776 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  RR )
2524recnd 9412 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  CC )
2625abscld 12922 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  e.  RR )
2711adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
2827, 23resubcld 9776 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  RR )
2928recnd 9412 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  CC )
3029abscld 12922 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  e.  RR )
31 rpre 10997 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
3231ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  x  e.  RR )
33 lelttr 9465 . . . . . . . . . 10  |-  ( ( ( abs `  (
( G `  k
)  -  A ) )  e.  RR  /\  ( abs `  ( ( F `  k )  -  A ) )  e.  RR  /\  x  e.  RR )  ->  (
( ( abs `  (
( G `  k
)  -  A ) )  <_  ( abs `  ( ( F `  k )  -  A
) )  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x )  ->  ( abs `  ( ( G `
 k )  -  A ) )  < 
x ) )
3426, 30, 32, 33syl3anc 1218 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( ( abs `  (
( G `  k
)  -  A ) )  <_  ( abs `  ( ( F `  k )  -  A
) )  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x )  ->  ( abs `  ( ( G `
 k )  -  A ) )  < 
x ) )
3521, 34mpand 675 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  -> 
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
369, 35sylan2 474 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3736anassrs 648 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3837ralimdva 2794 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( F `  k )  -  A ) )  <  x  ->  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3938reximdva 2828 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( ( G `  k )  -  A ) )  <  x ) )
408, 39mpd 15 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x )
4140ralrimiva 2799 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( ( G `  k )  -  A
) )  <  x
)
42 climsqz.5 . . 3  |-  ( ph  ->  G  e.  W )
43 eqidd 2444 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
4412recnd 9412 . . 3  |-  ( ph  ->  A  e.  CC )
4510recnd 9412 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
461, 2, 42, 43, 44, 45clim2c 12983 . 2  |-  ( ph  ->  ( G  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
4741, 46mpbird 232 1  |-  ( ph  ->  G  ~~>  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   RRcr 9281    < clt 9418    <_ cle 9419    - cmin 9595   ZZcz 10646   ZZ>=cuz 10861   RR+crp 10991   abscabs 12723    ~~> cli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fl 11642  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967
This theorem is referenced by:  expcnv  13326  explecnv  13327  plyeq0lem  21678  leibpi  22337  emcllem4  22392  basellem6  22423  basellem9  22426  lgamcvg2  27041  wallispilem5  29864  stirlinglem1  29869
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