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Theorem clim2c 11979
Description: Express the predicate  F converges to  A. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
clim2.1  |-  Z  =  ( ZZ>= `  M )
clim2.2  |-  ( ph  ->  M  e.  ZZ )
clim2.3  |-  ( ph  ->  F  e.  V )
clim2.4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
clim2c.5  |-  ( ph  ->  A  e.  CC )
clim2c.6  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
Assertion
Ref Expression
clim2c  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x )
)
Distinct variable groups:    j, k, x, A    j, F, k, x    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 clim2c
StepHypRef Expression
1 clim2c.5 . . 3  |-  ( ph  ->  A  e.  CC )
21biantrurd 494 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( B  e.  CC  /\  ( abs `  ( B  -  A ) )  <  x )  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) ) )
3 clim2.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
43uztrn2 10245 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
5 clim2c.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
65biantrurd 494 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
74, 6sylan2 460 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
87anassrs 629 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  -  A ) )  < 
x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
98ralbidva 2559 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  x  <->  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
109rexbidva 2560 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
1110ralbidv 2563 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( B  -  A ) )  <  x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) )
12 clim2.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
13 clim2.3 . . 3  |-  ( ph  ->  F  e.  V )
14 clim2.4 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
153, 12, 13, 14clim2 11978 . 2  |-  ( ph  ->  ( F  ~~>  A  <->  ( A  e.  CC  /\  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( B  e.  CC  /\  ( abs `  ( B  -  A )
)  <  x )
) ) )
162, 11, 153bitr4rd 277 1  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  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    < clt 8867    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  clim0c  11981  climconst  12017  rlimclim1  12019  2clim  12046  climcn1  12065  climcn2  12066  climsqz  12114  climsqz2  12115  climsup  12143  ulmclm  19766  itgulm  19784  climinf  27732
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-pre-lttri 8811  ax-pre-lttrn 8812
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-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-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-neg 9040  df-z 10025  df-uz 10231  df-clim 11962
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