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Theorem climsqz 10300
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 270 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  ZZ )
4 simpr 108 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5 eqidd 2083 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
6 climadd.4 . . . . . 6  |-  ( ph  ->  F  ~~>  A )
76adantr 270 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  F  ~~>  A )
81, 3, 4, 5, 7climi2 10254 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  x )
91uztrn2 8706 . . . . . . . 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 10289 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
1312adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
14 climsqz.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
1510, 11, 13, 14lesub2dd 7718 . . . . . . . . . . 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 10190 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  =  ( A  -  ( G `  k )
) )
1810, 11, 13, 14, 16letrd 7289 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  A )
1910, 13, 18abssuble0d 10190 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( A  -  ( F `  k )
) )
2015, 17, 193brtr4d 3817 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2120adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2211adantlr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
2312ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  A  e.  RR )
2422, 23resubcld 7541 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  RR )
2524recnd 7198 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  CC )
2625abscld 10194 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  e.  RR )
2710adantlr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
2827, 23resubcld 7541 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  RR )
2928recnd 7198 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  CC )
3029abscld 10194 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  e.  RR )
31 rpre 8810 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
3231ad2antlr 473 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  x  e.  RR )
33 lelttr 7255 . . . . . . . . . 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 1170 . . . . . . . . 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 420 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  -> 
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
369, 35sylan2 280 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3736anassrs 392 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3837ralimdva 2430 . . . . 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 2464 . . . 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 13 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x )
4140ralrimiva 2435 . 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 2083 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
4412recnd 7198 . . 3  |-  ( ph  ->  A  e.  CC )
4511recnd 7198 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
461, 2, 42, 43, 44, 45clim2c 10250 . 2  |-  ( ph  ->  ( G  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
4741, 46mpbird 165 1  |-  ( ph  ->  G  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   RRcr 7031    < clt 7204    <_ cle 7205    - cmin 7335   ZZcz 8421   ZZ>=cuz 8689   RR+crp 8804   abscabs 10010    ~~> cli 10244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145  ax-arch 7146  ax-caucvg 7147
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-if 3354  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-2 8154  df-3 8155  df-4 8156  df-n0 8345  df-z 8422  df-uz 8690  df-rp 8805  df-iseq 9511  df-iexp 9562  df-cj 9856  df-re 9857  df-im 9858  df-rsqrt 10011  df-abs 10012  df-clim 10245
This theorem is referenced by: (None)
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