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Theorem climabs0 11100
Description: Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climabs0.1  |-  Z  =  ( ZZ>= `  M )
climabs0.2  |-  ( ph  ->  M  e.  ZZ )
climabs0.3  |-  ( ph  ->  F  e.  V )
climabs0.4  |-  ( ph  ->  G  e.  W )
climabs0.5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climabs0.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
Assertion
Ref Expression
climabs0  |-  ( ph  ->  ( F  ~~>  0  <->  G  ~~>  0 ) )
Distinct variable groups:    k, F    k, G    ph, k    k, Z
Allowed substitution hints:    M( k)    V( k)    W( k)

Proof of Theorem climabs0
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climabs0.1 . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
21uztrn2 9362 . . . . . . 7  |-  ( ( j  e.  Z  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  Z )
3 climabs0.5 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
4 absidm 10894 . . . . . . . . 9  |-  ( ( F `  k )  e.  CC  ->  ( abs `  ( abs `  ( F `  k )
) )  =  ( abs `  ( F `
 k ) ) )
53, 4syl 14 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( abs `  ( F `  k )
) )  =  ( abs `  ( F `
 k ) ) )
65breq1d 3942 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
72, 6sylan2 284 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
87anassrs 397 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( abs `  ( F `  k )
) )  <  x  <->  ( abs `  ( F `
 k ) )  <  x ) )
98ralbidva 2433 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  ( A. k  e.  ( ZZ>=
`  j ) ( abs `  ( abs `  ( F `  k
) ) )  < 
x  <->  A. k  e.  (
ZZ>= `  j ) ( abs `  ( F `
 k ) )  <  x ) )
109rexbidva 2434 . . 3  |-  ( ph  ->  ( E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( F `  k )
)  <  x )
)
1110ralbidv 2437 . 2  |-  ( ph  ->  ( A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( abs `  ( F `  k
) ) )  < 
x  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j ) ( abs `  ( F `  k
) )  <  x
) )
12 climabs0.2 . . 3  |-  ( ph  ->  M  e.  ZZ )
13 climabs0.4 . . 3  |-  ( ph  ->  G  e.  W )
14 climabs0.6 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
153abscld 10977 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
1615recnd 7813 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  CC )
171, 12, 13, 14, 16clim0c 11079 . 2  |-  ( ph  ->  ( G  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( abs `  ( F `  k ) ) )  <  x ) )
18 climabs0.3 . . 3  |-  ( ph  ->  F  e.  V )
19 eqidd 2140 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
201, 12, 18, 19, 3clim0c 11079 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( F `  k )
)  <  x )
)
2111, 17, 203bitr4rd 220 1  |-  ( ph  ->  ( F  ~~>  0  <->  G  ~~>  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3932   ` cfv 5126   CCcc 7637   0cc0 7639    < clt 7819   ZZcz 9073   ZZ>=cuz 9345   RR+crp 9463   abscabs 10793    ~~> cli 11071
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756  ax-pre-mulext 7757  ax-arch 7758  ax-caucvg 7759
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-frec 6291  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-reap 8356  df-ap 8363  df-div 8452  df-inn 8740  df-2 8798  df-3 8799  df-4 8800  df-n0 8997  df-z 9074  df-uz 9346  df-rp 9464  df-seqfrec 10243  df-exp 10317  df-cj 10638  df-re 10639  df-im 10640  df-rsqrt 10794  df-abs 10795  df-clim 11072
This theorem is referenced by:  expcnvap0  11295  expcnv  11297  explecnv  11298
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