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Theorem climi0 12296
Description: Convergence of a sequence of complex numbers to zero. (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 )
climi0.5  |-  ( ph  ->  F  ~~>  0 )
Assertion
Ref Expression
climi0  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  B )  <  C )
Distinct variable groups:    j, k, C    j, F, k    ph, j,
k    j, Z, k    j, M
Allowed substitution hints:    B( j, k)    M( k)

Proof of Theorem climi0
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 climi0.5 . . 3  |-  ( ph  ->  F  ~~>  0 )
61, 2, 3, 4, 5climi 12294 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  C ) )
7 subid1 9312 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
87fveq2d 5724 . . . . . 6  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
98breq1d 4214 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  ( B  -  0 ) )  <  C  <->  ( abs `  B )  <  C
) )
109biimpa 471 . . . 4  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  C )  -> 
( abs `  B
)  <  C )
1110ralimi 2773 . . 3  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
C )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  C )
1211reximi 2805 . 2  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
C )  ->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  C )
136, 12syl 16 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  B )  <  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8978   0cc0 8980    < clt 9110    - cmin 9281   ZZcz 10272   ZZ>=cuz 10478   RR+crp 10602   abscabs 12029    ~~> cli 12268
This theorem is referenced by:  mertenslem2  12652  iscmet3lem3  19233  radcnvlem1  20319  abelthlem5  20341  abelthlem8  20345  sinccvg  25100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-z 10273  df-uz 10479  df-clim 12272
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