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Theorem clim0 11085
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 11083 . 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 7781 . . . 4  |-  0  e.  CC
76biantrur 301 . . 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 8005 . . . . . . . . 9  |-  ( B  e.  CC  ->  ( B  -  0 )  =  B )
98fveq2d 5432 . . . . . . . 8  |-  ( B  e.  CC  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
109breq1d 3946 . . . . . . 7  |-  ( B  e.  CC  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
1110pm5.32i 450 . . . . . 6  |-  ( ( B  e.  CC  /\  ( abs `  ( B  -  0 ) )  <  x )  <->  ( B  e.  CC  /\  ( abs `  B )  <  x
) )
1211ralbii 2444 . . . . 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 2445 . . . 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 2444 . . 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 185 . 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 195 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    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   CCcc 7641   0cc0 7643    < clt 7823    - cmin 7956   ZZcz 9077   ZZ>=cuz 9349   RR+crp 9469   abscabs 10800    ~~> cli 11078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-inn 8744  df-n0 9001  df-z 9078  df-uz 9350  df-clim 11079
This theorem is referenced by: (None)
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