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Theorem climsqz2 11244
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 274 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  M  e.  ZZ )
4 simpr 109 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5 eqidd 2158 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
6 climadd.4 . . . . . 6  |-  ( ph  ->  F  ~~>  A )
76adantr 274 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  F  ~~>  A )
81, 3, 4, 5, 7climi2 11196 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  (
( F `  k
)  -  A ) )  <  x )
91uztrn2 9461 . . . . . . . 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 11232 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  RR )
1312adantr 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  e.  RR )
14 climsqz2.8 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  <_  ( F `  k
) )
1510, 11, 13, 14lesub1dd 8440 . . . . . . . . . . 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 11087 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  =  ( ( G `  k )  -  A
) )
1813, 10, 11, 16, 14letrd 8003 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  A  <_  ( F `  k
) )
1913, 11, 18abssubge0d 11087 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( ( F `  k )  -  A
) )
2015, 17, 193brtr4d 3998 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2120adantlr 469 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  <_ 
( abs `  (
( F `  k
)  -  A ) ) )
2210adantlr 469 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
2312ad2antrr 480 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  A  e.  RR )
2422, 23resubcld 8260 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  RR )
2524recnd 7908 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( G `  k
)  -  A )  e.  CC )
2625abscld 11092 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( G `
 k )  -  A ) )  e.  RR )
2711adantlr 469 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
2827, 23resubcld 8260 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  RR )
2928recnd 7908 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  e.  CC )
3029abscld 11092 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  e.  RR )
31 rpre 9573 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  e.  RR )
3231ad2antlr 481 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  x  e.  RR )
33 lelttr 7968 . . . . . . . . . 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 1220 . . . . . . . . 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 426 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  -> 
( abs `  (
( G `  k
)  -  A ) )  <  x ) )
369, 35sylan2 284 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
j  e.  Z  /\  k  e.  ( ZZ>= `  j ) ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3736anassrs 398 . . . . . 6  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  j  e.  Z )  /\  k  e.  ( ZZ>=
`  j ) )  ->  ( ( abs `  ( ( F `  k )  -  A
) )  <  x  ->  ( abs `  (
( G `  k
)  -  A ) )  <  x ) )
3837ralimdva 2524 . . . . 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 2559 . . . 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 2530 . 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 2158 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( G `  k ) )
4412recnd 7908 . . 3  |-  ( ph  ->  A  e.  CC )
4510recnd 7908 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
461, 2, 42, 43, 44, 45clim2c 11192 . 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 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   class class class wbr 3967   ` cfv 5172  (class class class)co 5826   RRcr 7733    < clt 7914    <_ cle 7915    - cmin 8050   ZZcz 9172   ZZ>=cuz 9444   RR+crp 9566   abscabs 10908    ~~> cli 11186
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-rp 9567  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-clim 11187
This theorem is referenced by:  expcnvap0  11410  expcnvre  11411  explecnv  11413
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