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Theorem climabs0 12371
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 10495 . . . . . . 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 12119 . . . . . . . . 9  |-  ( ( F `  k )  e.  CC  ->  ( abs `  ( abs `  ( F `  k )
) )  =  ( abs `  ( F `
 k ) ) )
53, 4syl 16 . . . . . . . 8  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( abs `  ( F `  k )
) )  =  ( abs `  ( F `
 k ) ) )
65breq1d 4214 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
72, 6sylan2 461 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
87anassrs 630 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( abs `  ( F `  k )
) )  <  x  <->  ( abs `  ( F `
 k ) )  <  x ) )
98ralbidva 2713 . . . 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 2714 . . 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 2717 . 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 12230 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
1615recnd 9106 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  CC )
171, 12, 13, 14, 16clim0c 12293 . 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 2436 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
201, 12, 18, 19, 3clim0c 12293 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( F `  k )
)  <  x )
)
2111, 17, 203bitr4rd 278 1  |-  ( ph  ->  ( F  ~~>  0  <->  G  ~~>  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5446   CCcc 8980   0cc0 8982    < clt 9112   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   abscabs 12031    ~~> cli 12270
This theorem is referenced by:  expcnv  12635  explecnv  12636  plyeq0lem  20121  lgamcvg2  24831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274
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