Theorem expcnv 7233

Description: A sequence of powers of a complex number A with absolute value smaller than 1 converges to zero.

Hypothesis

Ref	Expression
expcnv.1	|- F e. V

Assertion

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

Distinct variable groups:   A,k   k,F

Proof of Theorem expcnv

Step	Hyp	Ref	Expression
1		expcnvlem6 7232	. . . . . . . 8 |- ((((abs` A) e. RR /\ 0 <_ (abs` A) /\ (abs` A) < 1) /\ (x e. RR /\ 0 < x)) -> E.j e. NN A.k e. NN (j <_ k -> ((abs` A)^k) < x))
2		absclt 6833	. . . . . . . . . 10 |- (A e. CC -> (abs` A) e. RR)
3	2	adantr 391	. . . . . . . . 9 |- ((A e. CC /\ (abs` A) < 1) -> (abs` A) e. RR)
4		absge0t 6854	. . . . . . . . . 10 |- (A e. CC -> 0 <_ (abs` A))
5	4	adantr 391	. . . . . . . . 9 |- ((A e. CC /\ (abs` A) < 1) -> 0 <_ (abs` A))
6		pm3.27 323	. . . . . . . . 9 |- ((A e. CC /\ (abs` A) < 1) -> (abs` A) < 1)
7	3, 5, 6	3jca 821	. . . . . . . 8 |- ((A e. CC /\ (abs` A) < 1) -> ((abs` A) e. RR /\ 0 <_ (abs` A) /\ (abs` A) < 1))
8	1, 7	sylan 450	. . . . . . 7 |- (((A e. CC /\ (abs` A) < 1) /\ (x e. RR /\ 0 < x)) -> E.j e. NN A.k e. NN (j <_ k -> ((abs` A)^k) < x))
9	8	adantllr 399	. . . . . 6 |- ((((A e. CC /\ A.k e. NN (F` k) = (A^k)) /\ (abs` A) < 1) /\ (x e. RR /\ 0 < x)) -> E.j e. NN A.k e. NN (j <_ k -> ((abs` A)^k) < x))
10		fveq2 3730	. . . . . . . . . . . . . . 15 |- ((F` k) = (A^k) -> (abs` (F` k)) = (abs` (A^k)))
11		absexpt 6868	. . . . . . . . . . . . . . . 16 |- ((A e. CC /\ k e. NN0) -> (abs` (A^k)) = ((abs` A)^k))
12		nnnn0t 6108	. . . . . . . . . . . . . . . 16 |- (k e. NN -> k e. NN0)
13	11, 12	sylan2 453	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ k e. NN) -> (abs` (A^k)) = ((abs` A)^k))
14	10, 13	sylan9eqr 1532	. . . . . . . . . . . . . 14 |- (((A e. CC /\ k e. NN) /\ (F` k) = (A^k)) -> (abs` (F` k)) = ((abs` A)^k))
15	14	breq1d 2634	. . . . . . . . . . . . 13 |- (((A e. CC /\ k e. NN) /\ (F` k) = (A^k)) -> ((abs` (F` k)) < x <-> ((abs` A)^k) < x))
16	15	imbi2d 614	. . . . . . . . . . . 12 |- (((A e. CC /\ k e. NN) /\ (F` k) = (A^k)) -> ((j <_ k -> (abs` (F` k)) < x) <-> (j <_ k -> ((abs` A)^k) < x)))
17	16	ex 373	. . . . . . . . . . 11 |- ((A e. CC /\ k e. NN) -> ((F` k) = (A^k) -> ((j <_ k -> (abs` (F` k)) < x) <-> (j <_ k -> ((abs` A)^k) < x))))
18	17	r19.20dva 1712	. . . . . . . . . 10 |- (A e. CC -> (A.k e. NN (F` k) = (A^k) -> A.k e. NN ((j <_ k -> (abs` (F` k)) < x) <-> (j <_ k -> ((abs` A)^k) < x))))
19	18	imp 350	. . . . . . . . 9 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k)) -> A.k e. NN ((j <_ k -> (abs` (F` k)) < x) <-> (j <_ k -> ((abs` A)^k) < x)))
20		r19.15 1756	. . . . . . . . 9 |- (A.k e. NN ((j <_ k -> (abs` (F` k)) < x) <-> (j <_ k -> ((abs` A)^k) < x)) -> (A.k e. NN (j <_ k -> (abs` (F` k)) < x) <-> A.k e. NN (j <_ k -> ((abs` A)^k) < x)))
21	19, 20	syl 10	. . . . . . . 8 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k)) -> (A.k e. NN (j <_ k -> (abs` (F` k)) < x) <-> A.k e. NN (j <_ k -> ((abs` A)^k) < x)))
22	21	rexbidv 1667	. . . . . . 7 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k)) -> (E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x) <-> E.j e. NN A.k e. NN (j <_ k -> ((abs` A)^k) < x)))
23	22	ad2antrr 406	. . . . . 6 |- ((((A e. CC /\ A.k e. NN (F` k) = (A^k)) /\ (abs` A) < 1) /\ (x e. RR /\ 0 < x)) -> (E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x) <-> E.j e. NN A.k e. NN (j <_ k -> ((abs` A)^k) < x)))
24	9, 23	mpbird 196	. . . . 5 |- ((((A e. CC /\ A.k e. NN (F` k) = (A^k)) /\ (abs` A) < 1) /\ (x e. RR /\ 0 < x)) -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))
25	24	exp32 379	. . . 4 |- (((A e. CC /\ A.k e. NN (F` k) = (A^k)) /\ (abs` A) < 1) -> (x e. RR -> (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))))
26	25	3impa 830	. . 3 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k) /\ (abs` A) < 1) -> (x e. RR -> (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))))
27	26	r19.21aiv 1716	. 2 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k) /\ (abs` A) < 1) -> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x)))
28		eleq1 1537	. . . . . . 7 |- ((F` k) = (A^k) -> ((F` k) e. CC <-> (A^k) e. CC))
29		expclt 6582	. . . . . . . 8 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
30	29, 12	sylan2 453	. . . . . . 7 |- ((A e. CC /\ k e. NN) -> (A^k) e. CC)
31	28, 30	syl5cbir 211	. . . . . 6 |- ((A e. CC /\ k e. NN) -> ((F` k) = (A^k) -> (F` k) e. CC))
32	31	r19.20dva 1712	. . . . 5 |- (A e. CC -> (A.k e. NN (F` k) = (A^k) -> A.k e. NN (F` k) e. CC))
33	32	imp 350	. . . 4 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k)) -> A.k e. NN (F` k) e. CC)
34		1z 6161	. . . . 5 |- 1 e. ZZ
35		nnuz 6440	. . . . . 6 |- NN = (ZZ>` 1)
36	35	eqimss2i 2115	. . . . 5 |- (ZZ>` 1) (_ NN
37		nnssz 6153	. . . . 5 |- NN (_ ZZ
38		expcnv.1	. . . . 5 |- F e. V
39	34, 36, 37, 34, 36, 37, 38	clm0 7083	. . . 4 |- (A.k e. NN (F` k) e. CC -> (F ~~> 0 <-> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))))
40	33, 39	syl 10	. . 3 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k)) -> (F ~~> 0 <-> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))))
41	40	3adant3 801	. 2 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k) /\ (abs` A) < 1) -> (F ~~> 0 <-> A.x e. RR (0 < x -> E.j e. NN A.k e. NN (j <_ k -> (abs` (F` k)) < x))))
42	27, 41	mpbird 196	1 |- ((A e. CC /\ A.k e. NN (F` k) = (A^k) /\ (abs` A) < 1) -> F ~~> 0)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  E.wrex 1649  Vcvv 1814   class class class wbr 2624  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246  1c1 5247   <_ cle 5307  NNcn 5308  NN0cn0 5309   < clt 5498  ZZ>cuz 6418  ^cexp 6569  abscabs 6751   ~~> cli 6974

This theorem is referenced by:  geolimilem 7235

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab<