Theorem explecnv 8495

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.

Hypothesis

Ref	Expression
expcnv.1	|- F e. _V

Assertion

Ref	Expression
explecnv	|- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> F ~~> 0)

Distinct variable groups:   A,k   k,F

Proof of Theorem explecnv

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . . . 8 |- (j = k -> (A^j) = (A^k))
2		eqid 1884	. . . . . . . 8 |- {<.j, x>. | (j e. NN /\ x = (A^j))} = {<.j, x>. | (j e. NN /\ x = (A^j))}
3		oprex 4907	. . . . . . . 8 |- (A^k) e. _V
4	1, 2, 3	fvopab4 4743	. . . . . . 7 |- (k e. NN -> ({<.j, x>. | (j e. NN /\ x = (A^j))}` k) = (A^k))
5	4	rgen 2159	. . . . . 6 |- A.k e. NN ({<.j, x>. | (j e. NN /\ x = (A^j))}` k) = (A^k)
6		nnex 7116	. . . . . . . 8 |- NN e. _V
7	6	opabex2 4539	. . . . . . 7 |- {<.j, x>. | (j e. NN /\ x = (A^j))} e. _V
8	7	expcnv 8494	. . . . . 6 |- ((A e. CC /\ A.k e. NN ({<.j, x>. | (j e. NN /\ x = (A^j))}` k) = (A^k) /\ (abs` A) < 1) -> {<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0)
9	5, 8	mp3an2 1179	. . . . 5 |- ((A e. CC /\ (abs` A) < 1) -> {<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0)
10		recn 6466	. . . . 5 |- (A e. RR -> A e. CC)
11	9, 10	sylan 497	. . . 4 |- ((A e. RR /\ (abs` A) < 1) -> {<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0)
12	11	3adant3 896	. . 3 |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> {<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0)
13		abscl 8084	. . . . . . . . . 10 |- ((F` k) e. CC -> (abs` (F` k)) e. RR)
14	13	adantr 425	. . . . . . . . 9 |- (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (abs` (F` k)) e. RR)
15	14	a1i 8	. . . . . . . 8 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (abs` (F` k)) e. RR))
16		reexpcl 7823	. . . . . . . . . 10 |- ((A e. RR /\ k e. NN0) -> (A^k) e. RR)
17		nnnn0 7315	. . . . . . . . . 10 |- (k e. NN -> k e. NN0)
18	16, 17	sylan2 500	. . . . . . . . 9 |- ((A e. RR /\ k e. NN) -> (A^k) e. RR)
19	18	adantlr 429	. . . . . . . 8 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (A^k) e. RR)
20	15, 19	jctild 662	. . . . . . 7 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> ((A^k) e. RR /\ (abs` (F` k)) e. RR)))
21		absge0 8105	. . . . . . . . . 10 |- ((F` k) e. CC -> 0 <_ (abs` (F` k)))
22	21	adantr 425	. . . . . . . . 9 |- (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> 0 <_ (abs` (F` k)))
23	22	a1i 8	. . . . . . . 8 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> 0 <_ (abs` (F` k))))
24		simpr 350	. . . . . . . . 9 |- (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (abs` (F` k)) <_ (A^k))
25	24	a1i 8	. . . . . . . 8 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (abs` (F` k)) <_ (A^k)))
26	23, 25	jcad 661	. . . . . . 7 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (0 <_ (abs` (F` k)) /\ (abs` (F` k)) <_ (A^k))))
27	20, 26	jcad 661	. . . . . 6 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (((A^k) e. RR /\ (abs` (F` k)) e. RR) /\ (0 <_ (abs` (F` k)) /\ (abs` (F` k)) <_ (A^k)))))
28	4	eleq1d 1963	. . . . . . . . 9 |- (k e. NN -> (({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR <-> (A^k) e. RR))
29		fveq2 4681	. . . . . . . . . . . 12 |- (j = k -> (F` j) = (F` k))
30	29	fveq2d 4685	. . . . . . . . . . 11 |- (j = k -> (abs` (F` j)) = (abs` (F` k)))
31		eqid 1884	. . . . . . . . . . 11 |- {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} = {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}
32		fvex 4689	. . . . . . . . . . 11 |- (abs` (F` k)) e. _V
33	30, 31, 32	fvopab4 4743	. . . . . . . . . 10 |- (k e. NN -> ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) = (abs` (F` k)))
34	33	eleq1d 1963	. . . . . . . . 9 |- (k e. NN -> (({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR <-> (abs` (F` k)) e. RR))
35	28, 34	anbi12d 690	. . . . . . . 8 |- (k e. NN -> ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) <-> ((A^k) e. RR /\ (abs` (F` k)) e. RR)))
36	33	breq2d 3350	. . . . . . . . 9 |- (k e. NN -> (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <-> 0 <_ (abs` (F` k))))
37	33, 4	breq12d 3351	. . . . . . . . 9 |- (k e. NN -> (({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k) <-> (abs` (F` k)) <_ (A^k)))
38	36, 37	anbi12d 690	. . . . . . . 8 |- (k e. NN -> ((0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)) <-> (0 <_ (abs` (F` k)) /\ (abs` (F` k)) <_ (A^k))))
39	35, 38	anbi12d 690	. . . . . . 7 |- (k e. NN -> (((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))) <-> (((A^k) e. RR /\ (abs` (F` k)) e. RR) /\ (0 <_ (abs` (F` k)) /\ (abs` (F` k)) <_ (A^k)))))
40	39	adantl 424	. . . . . 6 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))) <-> (((A^k) e. RR /\ (abs` (F` k)) e. RR) /\ (0 <_ (abs` (F` k)) /\ (abs` (F` k)) <_ (A^k)))))
41	27, 40	sylibrd 221	. . . . 5 |- (((A e. RR /\ (abs` A) < 1) /\ k e. NN) -> (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))))
42	41	ralimdvaa 2171	. . . 4 |- ((A e. RR /\ (abs` A) < 1) -> (A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> A.k e. NN ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))))
43	42	3impia 1064	. . 3 |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> A.k e. NN ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))))
44		1z 7368	. . . . 5 |- 1 e. ZZ
45	6	opabex2 4539	. . . . . 6 |- {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} e. _V
46		0cn 6481	. . . . . . 7 |- 0 e. CC
47	46	elisseti 2301	. . . . . 6 |- 0 e. _V
48	7, 45, 47	climsqueeze2 8401	. . . . 5 |- (({<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0 /\ 1 e. ZZ /\ A.k e. (ZZ>=` 1)((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))) -> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0)
49	44, 48	mp3an2 1179	. . . 4 |- (({<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0 /\ A.k e. (ZZ>=` 1)((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))) -> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0)
50		nnuz 7608	. . . . 5 |- NN = (ZZ>=` 1)
51		raleq 2266	. . . . 5 |- (NN = (ZZ>=` 1) -> (A.k e. NN ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))) <-> A.k e. (ZZ>=` 1)((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))))
52	50, 51	ax-mp 7	. . . 4 |- (A.k e. NN ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))) <-> A.k e. (ZZ>=` 1)((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k))))
53	49, 52	sylan2b 501	. . 3 |- (({<.j, x>. | (j e. NN /\ x = (A^j))} ~~> 0 /\ A.k e. NN ((({<.j, x>. | (j e. NN /\ x = (A^j))}` k) e. RR /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) e. RR) /\ (0 <_ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) /\ ({<.j, x>. | (j e. NN /\ x = (abs` (F` j)))}` k) <_ ({<.j, x>. | (j e. NN /\ x = (A^j))}` k)))) -> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0)
54	12, 43, 53	syl11anc 524	. 2 |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0)
55		simpl 346	. . . . 5 |- (((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (F` k) e. CC)
56	55	ralimi 2168	. . . 4 |- (A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> A.k e. NN (F` k) e. CC)
57		expcnv.1	. . . . 5 |- F e. _V
58	57, 45, 33	climabs0i 8373	. . . 4 |- (A.k e. NN (F` k) e. CC -> (F ~~> 0 <-> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0))
59	56, 58	syl 12	. . 3 |- (A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k)) -> (F ~~> 0 <-> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0))
60	59	3ad2ant3 899	. 2 |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> (F ~~> 0 <-> {<.j, x>. | (j e. NN /\ x = (abs` (F` j)))} ~~> 0))
61	54, 60	mpbird 213	1 |- ((A e. RR /\ (abs` A) < 1 /\ A.k e. NN ((F` k) e. CC /\ (abs` (F` k)) <_ (A^k))) -> F ~~> 0)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  ^cexp 7811  abscabs 8000   ~~> cli 8234

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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