Theorem climabs0 10146

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 8636	. . . . . . 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 9984	. . . . . . . . 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 3795	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
7	2, 6	sylan2 280	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
8	7	anassrs 392	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) )
9	8	ralbidva 2364	. . . 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 2365	. . 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 2368	. 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 10067	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
16	15	recnd 7147	. . 3 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC )
17	1, 12, 13, 14, 16	clim0c 10125	. 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 2082	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) )
20	1, 12, 18, 19, 3	clim0c 10125	. 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 219	1 |- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   class class class wbr 3785   ` cfv 4922   CCcc 6979   0cc0 6981   < clt 7153   ZZcz 8351   ZZ>=cuz 8619   RR+crp 8734   abscabs 9883   ~~> cli 10117

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885  df-clim 10118

This theorem is referenced by: (None)