Theorem climabs0 11489

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 9636	. . . . . . 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 11280	. . . . . . . . 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 4044	. . . . . . 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 2493	. . . 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 2494	. . 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 2497	. 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 11363	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
16	15	recnd 8072	. . 3 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC )
17	1, 12, 13, 14, 16	clim0c 11468	. 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 2197	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
20	1, 12, 18, 19, 3	clim0c 11468	. 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 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   class class class wbr 4034   ` cfv 5259   CCcc 7894   0cc0 7896   < clt 8078   ZZcz 9343   ZZ>=cuz 9618   RR+crp 9745   abscabs 11179   ~~> cli 11460

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461

This theorem is referenced by:  expcnvap0  11684  expcnv  11686  explecnv  11687