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Theorem clim2cf 31859
Description: Express the predicate  F converges to  A. Similar to clim2 13339, but without the disjoint var constraint  F k. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
clim2cf.nf  |-  F/_ k F
clim2cf.z  |-  Z  =  ( ZZ>= `  M )
clim2cf.m  |-  ( ph  ->  M  e.  ZZ )
clim2cf.f  |-  ( ph  ->  F  e.  V )
clim2cf.fv  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
clim2cf.a  |-  ( ph  ->  A  e.  CC )
clim2cf.b  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
Assertion
Ref Expression
clim2cf  |-  ( ph  ->  ( F  ~~>  A  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( B  -  A )
)  <  x )
)
Distinct variable groups:    A, j,
k, x    j, F, x    j, M    j, Z, k    ph, j, k, x
Allowed substitution hints:    B( x, j, k)    F( k)    M( x, k)    V( x, j, k)    Z( x)

Proof of Theorem clim2cf
StepHypRef Expression
1 clim2cf.a . . 3  |-  ( ph  ->  A  e.  CC )
21biantrurd 508 . 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 clim2cf.z . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
43uztrn2 11123 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
5 clim2cf.b . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  B  e.  CC )
65biantrurd 508 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
74, 6sylan2 474 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( B  -  A )
)  <  x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
87anassrs 648 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( B  -  A ) )  < 
x  <->  ( B  e.  CC  /\  ( abs `  ( B  -  A
) )  <  x
) ) )
98ralbidva 2893 . . . 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 2965 . . 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 2896 . 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 clim2cf.nf . . 3  |-  F/_ k F
13 clim2cf.m . . 3  |-  ( ph  ->  M  e.  ZZ )
14 clim2cf.f . . 3  |-  ( ph  ->  F  e.  V )
15 clim2cf.fv . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  B )
1612, 3, 13, 14, 15clim2f 31845 . 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 )
) ) )
172, 11, 163bitr4rd 286 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   F/_wnfc 2605   A.wral 2807   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507    < clt 9645    - cmin 9824   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   abscabs 13079    ~~> cli 13319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-neg 9827  df-z 10886  df-uz 11107  df-clim 13323
This theorem is referenced by:  clim0cf  31863
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