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Theorem climsqz 13220
Description: Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-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 )
climsqz.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
climsqz.9  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  A )
Assertion
Ref Expression
climsqz  |-  ( 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 climsqz
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 2452 . . . . 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 13091 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  x )
91uztrn2 10979 . . . . . . . 8  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
10 climsqz.6 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
11 climsqz.7 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
121, 2, 6, 10climrecl 13163 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
1312adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
14 climsqz.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
1510, 11, 13, 14lesub2dd 10057 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( A  -  ( G `  k ) )  <_ 
( A  -  ( F `  k )
) )
16 climsqz.9 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  A )
1711, 13, 16abssuble0d 13021 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  =  ( A  -  ( G `  k )
) )
1810, 11, 13, 14, 16letrd 9629 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  A )
1910, 13, 18abssuble0d 13021 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( A  -  ( F `  k )
) )
2015, 17, 193brtr4d 4420 . . . . . . . . . 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 ) ) )
2211adantlr 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 9877 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  RR )
2524recnd 9513 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  CC )
2625abscld 13024 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  e.  RR )
2710adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
2827, 23resubcld 9877 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  RR )
2928recnd 9513 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  CC )
3029abscld 13024 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  e.  RR )
31 rpre 11098 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
3231ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  x  e.  RR )
33 lelttr 9566 . . . . . . . . . 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 1219 . . . . . . . . 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 2824 . . . . 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 2924 . . . 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 2822 . 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 2452 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
4412recnd 9513 . . 3  |-  ( ph  ->  A  e.  CC )
4511recnd 9513 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
461, 2, 42, 43, 44, 45clim2c 13085 . 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   RRcr 9382    < clt 9519    <_ cle 9520    - cmin 9696   ZZcz 10747   ZZ>=cuz 10962   RR+crp 11092   abscabs 12825    ~~> cli 13064
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-rep 4501  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-fl 11743  df-seq 11908  df-exp 11967  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069
This theorem is referenced by:  supcvg  13420  mbfi1fseqlem6  21314  sinccvglem  27451
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