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Theorem clim2 12982
Description: Express the predicate: The limit of complex number sequence  F is  A, or  F converges to  A, with more general quantifier restrictions than clim 12972. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
clim2.1  |-  Z  =  ( ZZ>= `  M )
clim2.2  |-  ( ph  ->  M  e.  ZZ )
clim2.3  |-  ( ph  ->  F  e.  V )
clim2.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
Assertion
Ref Expression
clim2  |-  ( 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 )
) ) )
Distinct variable groups:    j, k, x, A    j, F, k, x    j, M    ph, j,
k, x    j, Z, k
Allowed substitution hints:    B( x, j, k)    M( x, k)    V( x, j, k)    Z( x)

Proof of Theorem clim2
StepHypRef Expression
1 clim2.3 . . 3  |-  ( ph  ->  F  e.  V )
2 eqidd 2444 . . 3  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( F `
 k )  =  ( F `  k
) )
31, 2clim 12972 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) ) )
4 clim2.1 . . . . . . . . . 10  |-  Z  =  ( ZZ>= `  M )
54uztrn2 10878 . . . . . . . . 9  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
6 clim2.4 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
76eleq1d 2509 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  CC  <->  B  e.  CC ) )
86oveq1d 6106 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  =  ( B  -  A ) )
98fveq2d 5695 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  ( B  -  A )
) )
109breq1d 4302 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  <->  ( abs `  ( B  -  A
) )  <  x
) )
117, 10anbi12d 710 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <-> 
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
125, 11sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <-> 
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
1312anassrs 648 . . . . . . 7  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( (
( F `  k
)  e.  CC  /\  ( abs `  ( ( F `  k )  -  A ) )  <  x )  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
1413ralbidva 2731 . . . . . 6  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( ( F `  k
)  e.  CC  /\  ( abs `  ( ( F `  k )  -  A ) )  <  x )  <->  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
1514rexbidva 2732 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <->  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  x ) ) )
16 clim2.2 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
174rexuz3 12836 . . . . . 6  |-  ( M  e.  ZZ  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( ( F `  k
)  e.  CC  /\  ( abs `  ( ( F `  k )  -  A ) )  <  x )  <->  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) )
1816, 17syl 16 . . . . 5  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <->  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x ) ) )
1915, 18bitr3d 255 . . . 4  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )  <->  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j ) ( ( F `  k )  e.  CC  /\  ( abs `  ( ( F `
 k )  -  A ) )  < 
x ) ) )
2019ralbidv 2735 . . 3  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) )
2120anbi2d 703 . 2  |-  ( ph  ->  ( ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
)  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  ZZ  A. k  e.  ( ZZ>= `  j )
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x ) ) ) )
223, 21bitr4d 256 1  |-  ( 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 )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   CCcc 9280    < clt 9418    - cmin 9595   ZZcz 10646   ZZ>=cuz 10861   RR+crp 10991   abscabs 12723    ~~> cli 12962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-neg 9598  df-z 10647  df-uz 10862  df-clim 12966
This theorem is referenced by:  clim2c  12983  clim0  12984  climi  12988  climrlim2  13025  climeq  13045  isercoll  13145  lmclim  20813
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