Theorem climabs0 11992

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 9872	. . . . . . 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 11783	. . . . . . . . 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 4119	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
7	2, 6	sylan2 286	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
8	7	anassrs 400	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
9	8	ralbidva 2538	. . . 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 2539	. . 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 2542	. 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 11866	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
16	15	recnd 8302	. . 3 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC )
17	1, 12, 13, 14, 16	clim0c 11971	. 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 2233	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
20	1, 12, 18, 19, 3	clim0c 11971	. 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 221	1 |- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   class class class wbr 4109   ` cfv 5352   CCcc 8125   0cc0 8127   < clt 8308   ZZcz 9577   ZZ>=cuz 9853   RR+crp 9986   abscabs 11682   ~~> cli 11963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964

This theorem is referenced by:  expcnvap0  12188  expcnv  12190  explecnv  12191