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

Theorem climi2 13111
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 13110 . 2  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
7 simpr 461 . . . 4  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C )  -> 
( abs `  ( B  -  A )
)  <  C )
87ralimi 2819 . . 3  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
C )  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  C )
98reximi 2929 . 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 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   CCcc 9395    < clt 9533    - cmin 9710   ZZcz 10761   ZZ>=cuz 10976   RR+crp 11106   abscabs 12845    ~~> cli 13084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-neg 9713  df-z 10762  df-uz 10977  df-clim 13088
This theorem is referenced by:  rlimclim  13146  climcn1  13191  climcn2  13192  climsqz  13240  climsqz2  13241  mertenslem2  13467  uniioombllem6  21211  ulmcau  22003  ulmdvlem3  22010  rrncmslem  28902  stoweidlem7  29973
  Copyright terms: Public domain W3C validator