MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  climi2 Structured version   Unicode version

Theorem climi2 12973
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 12972 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
7 simpr 458 . . . 4  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C )  -> 
( abs `  ( B  -  A )
)  <  C )
87ralimi 2781 . . 3  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
C )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  C )
98reximi 2813 . 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 16 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 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268    < clt 9406    - cmin 9583   ZZcz 10634   ZZ>=cuz 10849   RR+crp 10979   abscabs 12707    ~~> cli 12946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-pre-lttri 9344  ax-pre-lttrn 9345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-neg 9586  df-z 10635  df-uz 10850  df-clim 12950
This theorem is referenced by:  rlimclim  13008  climcn1  13053  climcn2  13054  climsqz  13102  climsqz2  13103  mertenslem2  13328  uniioombllem6  20910  ulmcau  21745  ulmdvlem3  21752  rrncmslem  28575  stoweidlem7  29648
  Copyright terms: Public domain W3C validator