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Mirrors > Home > ILE Home > Th. List > expcnvap0 | Unicode version |
Description: A sequence of powers of a complex number with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 23-Oct-2022.) |
Ref | Expression |
---|---|
expcnvap0.1 | |
expcnvap0.2 | |
expcnvap0.0 | # |
Ref | Expression |
---|---|
expcnvap0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9361 | . . 3 | |
2 | 1zzd 9081 | . . 3 | |
3 | expcnvap0.2 | . . . . . . . 8 | |
4 | expcnvap0.1 | . . . . . . . . . 10 | |
5 | expcnvap0.0 | . . . . . . . . . 10 # | |
6 | 4, 5 | absrpclapd 10960 | . . . . . . . . 9 |
7 | 6 | reclt1d 9497 | . . . . . . . 8 |
8 | 3, 7 | mpbid 146 | . . . . . . 7 |
9 | 1re 7765 | . . . . . . . 8 | |
10 | 6 | rpreccld 9494 | . . . . . . . . 9 |
11 | 10 | rpred 9483 | . . . . . . . 8 |
12 | difrp 9480 | . . . . . . . 8 | |
13 | 9, 11, 12 | sylancr 410 | . . . . . . 7 |
14 | 8, 13 | mpbid 146 | . . . . . 6 |
15 | 14 | rpreccld 9494 | . . . . 5 |
16 | 15 | rpcnd 9485 | . . . 4 |
17 | divcnv 11266 | . . . 4 | |
18 | 16, 17 | syl 14 | . . 3 |
19 | nnex 8726 | . . . . 5 | |
20 | 19 | mptex 5646 | . . . 4 |
21 | 20 | a1i 9 | . . 3 |
22 | simpr 109 | . . . . 5 | |
23 | 16 | adantr 274 | . . . . . 6 |
24 | 22 | nncnd 8734 | . . . . . 6 |
25 | 22 | nnap0d 8766 | . . . . . 6 # |
26 | 23, 24, 25 | divclapd 8550 | . . . . 5 |
27 | oveq2 5782 | . . . . . 6 | |
28 | eqid 2139 | . . . . . 6 | |
29 | 27, 28 | fvmptg 5497 | . . . . 5 |
30 | 22, 26, 29 | syl2anc 408 | . . . 4 |
31 | 15 | rpred 9483 | . . . . 5 |
32 | nndivre 8756 | . . . . 5 | |
33 | 31, 32 | sylan 281 | . . . 4 |
34 | 30, 33 | eqeltrd 2216 | . . 3 |
35 | 6 | adantr 274 | . . . . . . . 8 |
36 | 35 | rpcnd 9485 | . . . . . . 7 |
37 | nnnn0 8984 | . . . . . . . 8 | |
38 | 37 | adantl 275 | . . . . . . 7 |
39 | 36, 38 | expcld 10424 | . . . . . 6 |
40 | oveq2 5782 | . . . . . . 7 | |
41 | eqid 2139 | . . . . . . 7 | |
42 | 40, 41 | fvmptg 5497 | . . . . . 6 |
43 | 22, 39, 42 | syl2anc 408 | . . . . 5 |
44 | nnz 9073 | . . . . . 6 | |
45 | rpexpcl 10312 | . . . . . 6 | |
46 | 6, 44, 45 | syl2an 287 | . . . . 5 |
47 | 43, 46 | eqeltrd 2216 | . . . 4 |
48 | 47 | rpred 9483 | . . 3 |
49 | nnrp 9451 | . . . . . . 7 | |
50 | rpmulcl 9466 | . . . . . . 7 | |
51 | 14, 49, 50 | syl2an 287 | . . . . . 6 |
52 | 51 | rpred 9483 | . . . . . . . 8 |
53 | peano2re 7898 | . . . . . . . . 9 | |
54 | 52, 53 | syl 14 | . . . . . . . 8 |
55 | rpexpcl 10312 | . . . . . . . . . 10 | |
56 | 10, 44, 55 | syl2an 287 | . . . . . . . . 9 |
57 | 56 | rpred 9483 | . . . . . . . 8 |
58 | 52 | lep1d 8689 | . . . . . . . 8 |
59 | 11 | adantr 274 | . . . . . . . . 9 |
60 | 10 | rpge0d 9487 | . . . . . . . . . 10 |
61 | 60 | adantr 274 | . . . . . . . . 9 |
62 | bernneq2 10413 | . . . . . . . . 9 | |
63 | 59, 38, 61, 62 | syl3anc 1216 | . . . . . . . 8 |
64 | 52, 54, 57, 58, 63 | letrd 7886 | . . . . . . 7 |
65 | 6 | rpcnd 9485 | . . . . . . . 8 |
66 | 6 | rpap0d 9489 | . . . . . . . 8 # |
67 | exprecap 10334 | . . . . . . . 8 # | |
68 | 65, 66, 44, 67 | syl2an3an 1276 | . . . . . . 7 |
69 | 64, 68 | breqtrd 3954 | . . . . . 6 |
70 | 51, 46, 69 | lerec2d 9505 | . . . . 5 |
71 | 14 | rpcnd 9485 | . . . . . . 7 |
72 | 14 | rpap0d 9489 | . . . . . . 7 # |
73 | 71, 72 | jca 304 | . . . . . 6 # |
74 | nncn 8728 | . . . . . . 7 | |
75 | nnap0 8749 | . . . . . . 7 # | |
76 | 74, 75 | jca 304 | . . . . . 6 # |
77 | recdivap2 8485 | . . . . . 6 # # | |
78 | 73, 76, 77 | syl2an 287 | . . . . 5 |
79 | 70, 78 | breqtrrd 3956 | . . . 4 |
80 | 79, 43, 30 | 3brtr4d 3960 | . . 3 |
81 | 47 | rpge0d 9487 | . . 3 |
82 | 1, 2, 18, 21, 34, 48, 80, 81 | climsqz2 11105 | . 2 |
83 | nn0ex 8983 | . . . . 5 | |
84 | 83 | mptex 5646 | . . . 4 |
85 | 84 | a1i 9 | . . 3 |
86 | 4 | adantr 274 | . . . . . 6 |
87 | 86, 38 | expcld 10424 | . . . . 5 |
88 | oveq2 5782 | . . . . . 6 | |
89 | eqid 2139 | . . . . . 6 | |
90 | 88, 89 | fvmptg 5497 | . . . . 5 |
91 | 38, 87, 90 | syl2anc 408 | . . . 4 |
92 | expcl 10311 | . . . . 5 | |
93 | 4, 37, 92 | syl2an 287 | . . . 4 |
94 | 91, 93 | eqeltrd 2216 | . . 3 |
95 | absexp 10851 | . . . . 5 | |
96 | 4, 37, 95 | syl2an 287 | . . . 4 |
97 | 91 | fveq2d 5425 | . . . 4 |
98 | 96, 97, 43 | 3eqtr4rd 2183 | . . 3 |
99 | 1, 2, 85, 21, 94, 98 | climabs0 11076 | . 2 |
100 | 82, 99 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cvv 2686 class class class wbr 3929 cmpt 3989 cfv 5123 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 c1 7621 caddc 7623 cmul 7625 clt 7800 cle 7801 cmin 7933 # cap 8343 cdiv 8432 cn 8720 cn0 8977 cz 9054 crp 9441 cexp 10292 cabs 10769 cli 11047 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 |
This theorem is referenced by: expcnvre 11272 |
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