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Theorem clim0c 11981
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 0cn 8831 . . . 4  |-  0  e.  CC
65a1i 10 . . 3  |-  ( ph  ->  0  e.  CC )
7 clim0c.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
81, 2, 3, 4, 6, 7clim2c 11979 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  0 ) )  <  x ) )
91uztrn2 10245 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
107subid1d 9146 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( B  -  0 )  =  B )
1110fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
1211breq1d 4033 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
139, 12sylan2 460 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  0 ) )  <  x  <->  ( abs `  B )  <  x
) )
1413anassrs 629 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  - 
0 ) )  < 
x  <->  ( abs `  B
)  <  x )
)
1514ralbidva 2559 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  0 ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( abs `  B
)  <  x )
)
1615rexbidva 2560 . . 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 )
)
1716ralbidv 2563 . 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 )
)
188, 17bitrd 244 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  climabs0  12059  serf0  12153  iseralt  12157  lmclim2  26474  geomcau  26475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-z 10025  df-uz 10231  df-clim 11962
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