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Theorem climi0 11241
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 11239 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  C ) )
7 subid1 8128 . . . . . . 7  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
87fveq2d 5498 . . . . . 6  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
98breq1d 3997 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  ( B  -  0 ) )  <  C  <->  ( abs `  B )  <  C
) )
109biimpa 294 . . . 4  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  C )  -> 
( abs `  B
)  <  C )
1110ralimi 2533 . . 3  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
C )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  C )
1211reximi 2567 . 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 14 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 103    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3987   ` cfv 5196  (class class class)co 5851   CCcc 7761   0cc0 7763    < clt 7943    - cmin 8079   ZZcz 9201   ZZ>=cuz 9476   RR+crp 9599   abscabs 10950    ~~> cli 11230
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-addcom 7863  ax-addass 7865  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-0id 7871  ax-rnegex 7872  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-inn 8868  df-n0 9125  df-z 9202  df-uz 9477  df-clim 11231
This theorem is referenced by:  mertenslem2  11488
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