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Theorem clim2 10260
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 10258. (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 2083 . . 3  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( F `
 k )  =  ( F `  k
) )
31, 2clim 10258 . 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 8717 . . . . . . . . 9  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
6 clim2.4 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
76eleq1d 2148 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  e.  CC  <->  B  e.  CC ) )
86oveq1d 5558 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  Z )  ->  (
( F `  k
)  -  A )  =  ( B  -  A ) )
98fveq2d 5213 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( ( F `
 k )  -  A ) )  =  ( abs `  ( B  -  A )
) )
109breq1d 3803 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  (
( F `  k
)  -  A ) )  <  x  <->  ( abs `  ( B  -  A
) )  <  x
) )
117, 10anbi12d 457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  (
( ( F `  k )  e.  CC  /\  ( abs `  (
( F `  k
)  -  A ) )  <  x )  <-> 
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
125, 11sylan2 280 . . . . . . . 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 392 . . . . . . 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 2365 . . . . . 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 2366 . . . . 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 10014 . . . . . 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 14 . . . . 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 188 . . . 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 2369 . . 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 452 . 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 189 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    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   CCcc 7041    < clt 7215    - cmin 7346   ZZcz 8432   ZZ>=cuz 8700   RR+crp 8815   abscabs 10021    ~~> cli 10255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-clim 10256
This theorem is referenced by:  clim2c  10261  clim0  10262  climi  10264  climeq  10276
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