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Theorem climi 12224
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
climi  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( 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 climi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 climi.3 . 2  |-  ( ph  ->  C  e.  RR+ )
2 climi.5 . . . 4  |-  ( ph  ->  F  ~~>  A )
3 climi.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
4 climi.2 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
5 climrel 12206 . . . . . . 7  |-  Rel  ~~>
65brrelexi 4851 . . . . . 6  |-  ( F  ~~>  A  ->  F  e.  _V )
72, 6syl 16 . . . . 5  |-  ( ph  ->  F  e.  _V )
8 climi.4 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
93, 4, 7, 8clim2 12218 . . . 4  |-  ( ph  ->  ( F  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) ) )
102, 9mpbid 202 . . 3  |-  ( ph  ->  ( A  e.  CC  /\ 
A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
x ) ) )
1110simprd 450 . 2  |-  ( ph  ->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  < 
x ) )
12 breq2 4150 . . . . 5  |-  ( x  =  C  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( abs `  ( B  -  A
) )  <  C
) )
1312anbi2d 685 . . . 4  |-  ( x  =  C  ->  (
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) ) )
1413rexralbidv 2686 . . 3  |-  ( x  =  C  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  x )  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  C )
) )
1514rspcv 2984 . 2  |-  ( C  e.  RR+  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) ) )
161, 11, 15sylc 58 1  |-  ( ph  ->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   _Vcvv 2892   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914    < clt 9046    - cmin 9216   ZZcz 10207   ZZ>=cuz 10413   RR+crp 10537   abscabs 11959    ~~> cli 12198
This theorem is referenced by:  climi2  12225  climi0  12226  climuni  12266  2clim  12286  climcau  12384  caucvgb  12393  stoweidlem7  27417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-pre-lttri 8990  ax-pre-lttrn 8991
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-neg 9219  df-z 10208  df-uz 10414  df-clim 12202
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