Theorem explecnv 12146

Description: A sequence of terms converges to zero when it is less than powers of a number A whose absolute value is smaller than 1. (Contributed by NM, 19-Jul-2008.) (Revised by Mario Carneiro, 26-Apr-2014.)

Hypotheses

Ref	Expression
explecnv.1	|- Z = ( ZZ>= ` M )
explecnv.2	|- ( ph -> F e. V )
explecnv.3	|- ( ph -> M e. ZZ )
explecnv.5	|- ( ph -> A e. RR )
explecnv.4	|- ( ph -> ( abs ` A ) < 1 )
explecnv.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
explecnv.7	|- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) <_ ( A ^ k ) )

Assertion

Ref	Expression
explecnv	|- ( ph -> F ~~> 0 )

Distinct variable groups:   A, k   ph, k   k, F   k, Z   k, M

Allowed substitution hint:   V( k)

Proof of Theorem explecnv

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2231	. . 3 |- ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) = ( ZZ>= ` if ( M <_ 0 , 0 , M ) )
2		0z 9551	. . . 4 |- 0 e. ZZ
3		explecnv.3	. . . 4 |- ( ph -> M e. ZZ )
4		0zd 9552	. . . . 5 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> 0 e. ZZ )
5		simpr 110	. . . . 5 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> M e. ZZ )
6		zdcle 9617	. . . . . 6 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M <_ 0 )
7	6	ancoms 268	. . . . 5 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> DECID M <_ 0 )
8	4, 5, 7	ifcldcd 3647	. . . 4 |- ( ( 0 e. ZZ /\ M e. ZZ ) -> if ( M <_ 0 , 0 , M ) e. ZZ )
9	2, 3, 8	sylancr 414	. . 3 |- ( ph -> if ( M <_ 0 , 0 , M ) e. ZZ )
10		explecnv.5	. . . . 5 |- ( ph -> A e. RR )
11	10	recnd 8267	. . . 4 |- ( ph -> A e. CC )
12		explecnv.4	. . . 4 |- ( ph -> ( abs ` A ) < 1 )
13	11, 12	expcnv 12145	. . 3 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
14		zex 9549	. . . . . 6 |- ZZ e. _V
15		explecnv.1	. . . . . . 7 |- Z = ( ZZ>= ` M )
16		uzssz 9837	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
17	15, 16	eqsstri 3260	. . . . . 6 |- Z C_ ZZ
18	14, 17	ssexi 4232	. . . . 5 |- Z e. _V
19	18	mptex 5890	. . . 4 |- ( n e. Z |-> ( abs ` ( F ` n ) ) ) e. _V
20	19	a1i 9	. . 3 |- ( ph -> ( n e. Z |-> ( abs ` ( F ` n ) ) ) e. _V )
21		nn0uz 9852	. . . . . . . . . 10 |- NN0 = ( ZZ>= ` 0 )
22	15, 21	ineq12i 3408	. . . . . . . . 9 |- ( Z i^i NN0 ) = ( ( ZZ>= ` M ) i^i ( ZZ>= ` 0 ) )
23		uzin 9850	. . . . . . . . . 10 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( ( ZZ>= ` M ) i^i ( ZZ>= ` 0 ) ) = ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) )
24	3, 2, 23	sylancl 413	. . . . . . . . 9 |- ( ph -> ( ( ZZ>= ` M ) i^i ( ZZ>= ` 0 ) ) = ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) )
25	22, 24	eqtr2id 2277	. . . . . . . 8 |- ( ph -> ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) = ( Z i^i NN0 ) )
26	25	eleq2d 2301	. . . . . . 7 |- ( ph -> ( k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) <-> k e. ( Z i^i NN0 ) ) )
27	26	biimpa 296	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> k e. ( Z i^i NN0 ) )
28	27	elin2d 3399	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> k e. NN0 )
29	11	adantr 276	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> A e. CC )
30	29, 28	expcld 10998	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( A ^ k ) e. CC )
31		oveq2 6036	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
32		eqid 2231	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
33	31, 32	fvmptg 5731	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
34	28, 30, 33	syl2anc 411	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
35	10	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> A e. RR )
36	35, 28	reexpcld 11015	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( A ^ k ) e. RR )
37	34, 36	eqeltrd 2308	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. RR )
38	27	elin1d 3398	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> k e. Z )
39		explecnv.6	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
40	38, 39	syldan 282	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( F ` k ) e. CC )
41	40	abscld 11821	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( abs ` ( F ` k ) ) e. RR )
42		2fveq3 5653	. . . . . 6 |- ( n = k -> ( abs ` ( F ` n ) ) = ( abs ` ( F ` k ) ) )
43		eqid 2231	. . . . . 6 |- ( n e. Z |-> ( abs ` ( F ` n ) ) ) = ( n e. Z |-> ( abs ` ( F ` n ) ) )
44	42, 43	fvmptg 5731	. . . . 5 |- ( ( k e. Z /\ ( abs ` ( F ` k ) ) e. RR ) -> ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) = ( abs ` ( F ` k ) ) )
45	38, 41, 44	syl2anc 411	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) = ( abs ` ( F ` k ) ) )
46	45, 41	eqeltrd 2308	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) e. RR )
47		explecnv.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) <_ ( A ^ k ) )
48	38, 47	syldan 282	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( abs ` ( F ` k ) ) <_ ( A ^ k ) )
49	48, 45, 34	3brtr4d 4125	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) <_ ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) )
50	40	absge0d 11824	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> 0 <_ ( abs ` ( F ` k ) ) )
51	50, 45	breqtrrd 4121	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` if ( M <_ 0 , 0 , M ) ) ) -> 0 <_ ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) )
52	1, 9, 13, 20, 37, 46, 49, 51	climsqz2 11976	. 2 |- ( ph -> ( n e. Z |-> ( abs ` ( F ` n ) ) ) ~~> 0 )
53		explecnv.2	. . 3 |- ( ph -> F e. V )
54		simpr 110	. . . 4 |- ( ( ph /\ k e. Z ) -> k e. Z )
55	39	abscld 11821	. . . 4 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
56	54, 55, 44	syl2anc 411	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( n e. Z |-> ( abs ` ( F ` n ) ) ) ` k ) = ( abs ` ( F ` k ) ) )
57	15, 3, 53, 20, 39, 56	climabs0 11947	. 2 |- ( ph -> ( F ~~> 0 <-> ( n e. Z |-> ( abs ` ( F ` n ) ) ) ~~> 0 ) )
58	52, 57	mpbird 167	1 |- ( ph -> F ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   ifcif 3607   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   < clt 8273   <_ cle 8274   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   ^cexp 10863   abscabs 11637   ~~> cli 11918

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919

This theorem is referenced by: (None)