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Theorem climabs0 10284
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 8717 . . . . . . 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 10122 . . . . . . . . 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 3803 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
72, 6sylan2 280 . . . . . 6  |-  ( (
ph  /\  ( j  e.  Z  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( abs `  ( abs `  ( F `  k ) ) )  <  x  <->  ( abs `  ( F `  k
) )  <  x
) )
87anassrs 392 . . . . 5  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( ( abs `  ( abs `  ( F `  k )
) )  <  x  <->  ( abs `  ( F `
 k ) )  <  x ) )
98ralbidva 2365 . . . 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 2366 . . 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 2369 . 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 10205 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
1615recnd 7209 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  CC )
171, 12, 13, 14, 16clim0c 10263 . 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 2083 . . 3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  =  ( F `  k ) )
201, 12, 18, 19, 3clim0c 10263 . 2  |-  ( ph  ->  ( F  ~~>  0  <->  A. x  e.  RR+  E. j  e.  Z  A. k  e.  ( ZZ>= `  j )
( abs `  ( F `  k )
)  <  x )
)
2111, 17, 203bitr4rd 219 1  |-  ( ph  ->  ( F  ~~>  0  <->  G  ~~>  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   class class class wbr 3793   ` cfv 4932   CCcc 7041   0cc0 7043    < clt 7215   ZZcz 8432   ZZ>=cuz 8700   RR+crp 8815   abscabs 10021    ~~> cli 10255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023  df-clim 10256
This theorem is referenced by: (None)
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