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Theorem clim0c 11023
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 7727 . . 3  |-  ( ph  ->  0  e.  CC )
6 clim0c.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
71, 2, 3, 4, 5, 6clim2c 11021 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  0 ) )  <  x ) )
81uztrn2 9311 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
96subid1d 8030 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  -  0 )  =  B )
109fveq2d 5393 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
1110breq1d 3909 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
128, 11sylan2 284 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
1312anassrs 397 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  - 
0 ) )  < 
x  <->  ( abs `  B
)  <  x )
)
1413ralbidva 2410 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  0 ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  x )
)
1514rexbidva 2411 . . 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 2414 . 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 187 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   CCcc 7586   0cc0 7588    < clt 7768    - cmin 7901   ZZcz 9022   ZZ>=cuz 9294   RR+crp 9409   abscabs 10737    ~~> cli 11015
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023  df-uz 9295  df-clim 11016
This theorem is referenced by:  climabs0  11044  serf0  11089  divcnv  11234
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