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Theorem climi 12296
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 12278 . . . . . . 7  |-  Rel  ~~>
65brrelexi 4910 . . . . . 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 12290 . . . 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 4208 . . . . 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 2741 . . 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 3040 . 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 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980    < clt 9112    - cmin 9283   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   abscabs 12031    ~~> cli 12270
This theorem is referenced by:  climi2  12297  climi0  12298  climuni  12338  2clim  12358  climcau  12456  caucvgb  12465  stoweidlem7  27723
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 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
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-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-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-neg 9286  df-z 10275  df-uz 10481  df-clim 12274
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