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Theorem climi2 13483
Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climi.1  |-  Z  =  ( ZZ>= `  M )
climi.2  |-  ( ph  ->  M  e.  ZZ )
climi.3  |-  ( ph  ->  C  e.  RR+ )
climi.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
climi.5  |-  ( ph  ->  F  ~~>  A )
Assertion
Ref Expression
climi2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  C )
Distinct variable groups:    j, k, A    C, j, k    j, F, k    ph, j, k   
j, Z, k    j, M
Allowed substitution hints:    B( j, k)    M( k)

Proof of Theorem climi2
StepHypRef Expression
1 climi.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 climi.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 climi.3 . . 3  |-  ( ph  ->  C  e.  RR+ )
4 climi.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
5 climi.5 . . 3  |-  ( ph  ->  F  ~~>  A )
61, 2, 3, 4, 5climi 13482 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
7 simpr 459 . . . 4  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C )  -> 
( abs `  ( B  -  A )
)  <  C )
87ralimi 2797 . . 3  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
C )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  C )
98reximi 2872 . 2  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
C )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  C )
106, 9syl 17 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   CCcc 9520    < clt 9658    - cmin 9841   ZZcz 10905   ZZ>=cuz 11127   RR+crp 11265   abscabs 13216    ~~> cli 13456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-neg 9844  df-z 10906  df-uz 11128  df-clim 13460
This theorem is referenced by:  rlimclim  13518  climcn1  13563  climcn2  13564  climsqz  13612  climsqz2  13613  mertenslem2  13846  uniioombllem6  22289  ulmcau  23082  ulmdvlem3  23089  rrncmslem  31610  cvgdvgrat  36042  stoweidlem7  37157
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