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Theorem climabs0 13174
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 10982 . . . . . . 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 12922 . . . . . . . . 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 4403 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
72, 6sylan2 474 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
87anassrs 648 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( abs `  ( F `  k )
) )  <  x  <->  ( abs `  ( F `
 k ) )  <  x ) )
98ralbidva 2839 . . . 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 2852 . . 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 2841 . 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 13033 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
1615recnd 9516 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  CC )
171, 12, 13, 14, 16clim0c 13096 . 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 2452 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
201, 12, 18, 19, 3clim0c 13096 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( F `  k )
)  <  x )
)
2111, 17, 203bitr4rd 286 1  |-  ( ph  ->  ( F  ~~>  0  <->  G  ~~>  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   class class class wbr 4393   ` cfv 5519   CCcc 9384   0cc0 9386    < clt 9522   ZZcz 10750   ZZ>=cuz 10965   RR+crp 11095   abscabs 12834    ~~> cli 13073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077
This theorem is referenced by:  expcnv  13437  explecnv  13438  plyeq0lem  21804  lgamcvg2  27178
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