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Theorem clim0c 10252
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 )
clim0c.6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
Assertion
Ref Expression
clim0c  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( 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 clim0c
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 )
5 0cnd 7163 . . 3  |-  ( ph  ->  0  e.  CC )
6 clim0c.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
71, 2, 3, 4, 5, 6clim2c 10250 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  0 ) )  <  x ) )
81uztrn2 8706 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
96subid1d 7464 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  -  0 )  =  B )
109fveq2d 5207 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
1110breq1d 3797 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
128, 11sylan2 280 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
1312anassrs 392 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  - 
0 ) )  < 
x  <->  ( abs `  B
)  <  x )
)
1413ralbidva 2365 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  0 ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  x )
)
1514rexbidva 2366 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  0 ) )  <  x  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  x )
)
1615ralbidv 2369 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  0 ) )  <  x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  x )
)
177, 16bitrd 186 1  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  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 3787   ` cfv 4926  (class class class)co 5537   CCcc 7030   0cc0 7032    < clt 7204    - cmin 7335   ZZcz 8421   ZZ>=cuz 8689   RR+crp 8804   abscabs 10010    ~~> cli 10244
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 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143
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 3354  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-inn 8096  df-n0 8345  df-z 8422  df-uz 8690  df-clim 10245
This theorem is referenced by:  climabs0  10273  serif0  10316
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