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Theorem clim2 12225
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 12215. (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 2388 . . 3  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( F `
 k )  =  ( F `  k
) )
31, 2clim 12215 . 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 10435 . . . . . . . . 9  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
6 clim2.4 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
76eleq1d 2453 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  CC  <->  B  e.  CC ) )
86oveq1d 6035 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  =  ( B  -  A ) )
98fveq2d 5672 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  ( B  -  A )
) )
109breq1d 4163 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  <->  ( abs `  ( B  -  A
) )  <  x
) )
117, 10anbi12d 692 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <-> 
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
125, 11sylan2 461 . . . . . . . 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 630 . . . . . . 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 2665 . . . . . 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 2666 . . . . 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 12079 . . . . . 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 247 . . . 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 2669 . . 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 685 . 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 248 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   CCcc 8921    < clt 9053    - cmin 9223   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   abscabs 11966    ~~> cli 12205
This theorem is referenced by:  clim2c  12226  clim0  12227  climi  12231  climrlim2  12268  climeq  12288  isercoll  12388  lmclim  19126
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-pre-lttri 8997  ax-pre-lttrn 8998
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-neg 9226  df-z 10215  df-uz 10421  df-clim 12209
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