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Theorem clim0 12227
Description: Express the predicate  F converges to  0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
clim0.1  |-  Z  =  ( ZZ>= `  M )
clim0.2  |-  ( ph  ->  M  e.  ZZ )
clim0.3  |-  ( ph  ->  F  e.  V )
clim0.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
Assertion
Ref Expression
clim0  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  B
)  <  x )
) )
Distinct variable groups:    j, k, x, F    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 clim0
StepHypRef Expression
1 clim0.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 clim0.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 clim0.3 . . 3  |-  ( ph  ->  F  e.  V )
4 clim0.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
51, 2, 3, 4clim2 12225 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  (
0  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x ) ) ) )
6 0cn 9017 . . . 4  |-  0  e.  CC
76biantrur 493 . . 3  |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x )  <-> 
( 0  e.  CC  /\ 
A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
x ) ) )
8 subid1 9254 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
98fveq2d 5672 . . . . . . . 8  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
109breq1d 4163 . . . . . . 7  |-  ( B  e.  CC  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
1110pm5.32i 619 . . . . . 6  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x )  <->  ( B  e.  CC  /\  ( abs `  B )  <  x
) )
1211ralbii 2673 . . . . 5  |-  ( A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
x )  <->  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  B
)  <  x )
)
1312rexbii 2674 . . . 4  |-  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  ( B  - 
0 ) )  < 
x )  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  B
)  <  x )
)
1413ralbii 2673 . . 3  |-  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  B
)  <  x )
)
157, 14bitr3i 243 . 2  |-  ( ( 0  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x ) )  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( B  e.  CC  /\  ( abs `  B )  < 
x ) )
165, 15syl6bb 253 1  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  B
)  <  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   0cc0 8923    < clt 9053    - cmin 9223   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   abscabs 11966    ~~> cli 12205
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-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
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-reu 2656  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-oprab 6024  df-mpt2 6025  df-riota 6485  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-sub 9225  df-neg 9226  df-z 10215  df-uz 10421  df-clim 12209
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