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Theorem climsqz2 11056
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 272 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  ZZ )
4 simpr 109 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5 eqidd 2116 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
6 climadd.4 . . . . . 6  |-  ( ph  ->  F  ~~>  A )
76adantr 272 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  F  ~~>  A )
81, 3, 4, 5, 7climi2 11008 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  x )
91uztrn2 9295 . . . . . . . 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 11044 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
1312adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
14 climsqz2.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k
) )
1510, 11, 13, 14lesub1dd 8286 . . . . . . . . . . 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 10899 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  =  ( ( G `  k )  -  A
) )
1813, 10, 11, 16, 14letrd 7850 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
) )
1913, 11, 18abssubge0d 10899 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( ( F `  k )  -  A
) )
2015, 17, 193brtr4d 3928 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2120adantlr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2210adantlr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
2312ad2antrr 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  A  e.  RR )
2422, 23resubcld 8107 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  RR )
2524recnd 7758 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  CC )
2625abscld 10904 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  e.  RR )
2711adantlr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
2827, 23resubcld 8107 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  RR )
2928recnd 7758 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  CC )
3029abscld 10904 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  e.  RR )
31 rpre 9399 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
3231ad2antlr 478 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  x  e.  RR )
33 lelttr 7816 . . . . . . . . . 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 1199 . . . . . . . . 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 423 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  -> 
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
369, 35sylan2 282 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3736anassrs 395 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3837ralimdva 2474 . . . . 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 2509 . . . 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 2480 . 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 2116 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
4412recnd 7758 . . 3  |-  ( ph  ->  A  e.  CC )
4510recnd 7758 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
461, 2, 42, 43, 44, 45clim2c 11004 . 2  |-  ( ph  ->  ( G  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
4741, 46mpbird 166 1  |-  ( ph  ->  G  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   RRcr 7583    < clt 7764    <_ cle 7765    - cmin 7897   ZZcz 9008   ZZ>=cuz 9278   RR+crp 9393   abscabs 10720    ~~> cli 10998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-rp 9394  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-clim 10999
This theorem is referenced by:  expcnvap0  11222  expcnvre  11223  explecnv  11225
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