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Theorem climabs0 13490
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 11099 . . . . . . 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 13238 . . . . . . . . 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 4449 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
72, 6sylan2 472 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
87anassrs 646 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( abs `  ( F `  k )
) )  <  x  <->  ( abs `  ( F `
 k ) )  <  x ) )
98ralbidva 2890 . . . 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 2962 . . 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 2893 . 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 13349 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
1615recnd 9611 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  CC )
171, 12, 13, 14, 16clim0c 13412 . 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 2455 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
201, 12, 18, 19, 3clim0c 13412 . 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   class class class wbr 4439   ` cfv 5570   CCcc 9479   0cc0 9481    < clt 9617   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   abscabs 13149    ~~> cli 13389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393
This theorem is referenced by:  expcnv  13757  explecnv  13758  plyeq0lem  22773  lgamcvg2  28861
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