Theorem climabs0 11076

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.

Step	Hyp	Ref	Expression
1		climabs0.1	. . . . . . . 8 |- Z = ( ZZ>= ` M )
2	1	uztrn2 9343	. . . . . . 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 10870	. . . . . . . . 9 |- ( ( F ` k ) e. CC -> ( abs ` ( abs ` ( F ` k ) ) ) = ( abs ` ( F ` k ) ) )
5	3, 4	syl 14	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( abs ` ( abs ` ( F ` k ) ) ) = ( abs ` ( F ` k ) ) )
6	5	breq1d 3939	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
7	2, 6	sylan2 284	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
8	7	anassrs 397	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
9	8	ralbidva 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 ) )
10	9	rexbidva 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 ) )
11	10	ralbidv 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 ) ) )
15	3	abscld 10953	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
16	15	recnd 7794	. . 3 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC )
17	1, 12, 13, 14, 16	clim0c 11055	. 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 ) )
20	1, 12, 18, 19, 3	clim0c 11055	. 2 |- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < x ) )
21	11, 17, 20	3bitr4rd 220	1 |- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   ` cfv 5123   CCcc 7618   0cc0 7620   < clt 7800   ZZcz 9054   ZZ>=cuz 9326   RR+crp 9441   abscabs 10769   ~~> cli 11047

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 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

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 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-rp 9442  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048

This theorem is referenced by:  expcnvap0  11271  expcnv  11273  explecnv  11274