Theorem climabs0 11270

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 9504	. . . . . . 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 11062	. . . . . . . . 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 3999	. . . . . . 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 398	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
9	8	ralbidva 2466	. . . 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 2467	. . 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 2470	. 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 11145	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
16	15	recnd 7948	. . 3 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC )
17	1, 12, 13, 14, 16	clim0c 11249	. 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 2171	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
20	1, 12, 18, 19, 3	clim0c 11249	. 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 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   ` cfv 5198   CCcc 7772   0cc0 7774   < clt 7954   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961   ~~> cli 11241

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242

This theorem is referenced by:  expcnvap0  11465  expcnv  11467  explecnv  11468