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Mirrors > Home > ILE Home > Th. List > resqrexlemcvg | Unicode version |
Description: Lemma for resqrex 10962. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | |
resqrexlemex.a | |
resqrexlemex.agt0 |
Ref | Expression |
---|---|
resqrexlemcvg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrexlemex.seq | . . . 4 | |
2 | resqrexlemex.a | . . . 4 | |
3 | resqrexlemex.agt0 | . . . 4 | |
4 | 1, 2, 3 | resqrexlemf 10943 | . . 3 |
5 | rpssre 9594 | . . . 4 | |
6 | 5 | a1i 9 | . . 3 |
7 | 4, 6 | fssd 5347 | . 2 |
8 | 1nn 8862 | . . . . . . 7 | |
9 | 8 | a1i 9 | . . . . . 6 |
10 | 4, 9 | ffvelrnd 5618 | . . . . 5 |
11 | 2z 9213 | . . . . . 6 | |
12 | 11 | a1i 9 | . . . . 5 |
13 | 10, 12 | rpexpcld 10606 | . . . 4 |
14 | 2rp 9588 | . . . . 5 | |
15 | 14 | a1i 9 | . . . 4 |
16 | 13, 15 | rpmulcld 9643 | . . 3 |
17 | 16, 15 | rpmulcld 9643 | . 2 |
18 | 4 | ad2antrr 480 | . . . . . . . . . 10 |
19 | simplr 520 | . . . . . . . . . 10 | |
20 | 18, 19 | ffvelrnd 5618 | . . . . . . . . 9 |
21 | 20 | rpred 9626 | . . . . . . . 8 |
22 | eluznn 9532 | . . . . . . . . . . 11 | |
23 | 22 | adantll 468 | . . . . . . . . . 10 |
24 | 18, 23 | ffvelrnd 5618 | . . . . . . . . 9 |
25 | 24 | rpred 9626 | . . . . . . . 8 |
26 | 21, 25 | resubcld 8273 | . . . . . . 7 |
27 | 17 | ad2antrr 480 | . . . . . . . . 9 |
28 | 14 | a1i 9 | . . . . . . . . . 10 |
29 | 19 | nnzd 9306 | . . . . . . . . . 10 |
30 | 28, 29 | rpexpcld 10606 | . . . . . . . . 9 |
31 | 27, 30 | rpdivcld 9644 | . . . . . . . 8 |
32 | 31 | rpred 9626 | . . . . . . 7 |
33 | 19 | nnrpd 9624 | . . . . . . . . 9 |
34 | 27, 33 | rpdivcld 9644 | . . . . . . . 8 |
35 | 34 | rpred 9626 | . . . . . . 7 |
36 | 2 | ad2antrr 480 | . . . . . . . . 9 |
37 | 3 | ad2antrr 480 | . . . . . . . . 9 |
38 | eluzle 9472 | . . . . . . . . . 10 | |
39 | 38 | adantl 275 | . . . . . . . . 9 |
40 | 1, 36, 37, 19, 23, 39 | resqrexlemnm 10954 | . . . . . . . 8 |
41 | 2cn 8922 | . . . . . . . . . . 11 | |
42 | expm1t 10477 | . . . . . . . . . . 11 | |
43 | 41, 19, 42 | sylancr 411 | . . . . . . . . . 10 |
44 | 43 | oveq2d 5855 | . . . . . . . . 9 |
45 | 8 | a1i 9 | . . . . . . . . . . . . . 14 |
46 | 18, 45 | ffvelrnd 5618 | . . . . . . . . . . . . 13 |
47 | 11 | a1i 9 | . . . . . . . . . . . . 13 |
48 | 46, 47 | rpexpcld 10606 | . . . . . . . . . . . 12 |
49 | 48, 28 | rpmulcld 9643 | . . . . . . . . . . 11 |
50 | 49 | rpcnd 9628 | . . . . . . . . . 10 |
51 | 41 | a1i 9 | . . . . . . . . . . 11 |
52 | nnm1nn0 9149 | . . . . . . . . . . . 12 | |
53 | 19, 52 | syl 14 | . . . . . . . . . . 11 |
54 | 51, 53 | expcld 10582 | . . . . . . . . . 10 |
55 | 2ap0 8944 | . . . . . . . . . . . 12 # | |
56 | 55 | a1i 9 | . . . . . . . . . . 11 # |
57 | 1zzd 9212 | . . . . . . . . . . . 12 | |
58 | 29, 57 | zsubcld 9312 | . . . . . . . . . . 11 |
59 | 51, 56, 58 | expap0d 10588 | . . . . . . . . . 10 # |
60 | 50, 54, 51, 59, 56 | divcanap5rd 8708 | . . . . . . . . 9 |
61 | 44, 60 | eqtrd 2197 | . . . . . . . 8 |
62 | 40, 61 | breqtrrd 4007 | . . . . . . 7 |
63 | uzid 9474 | . . . . . . . . . 10 | |
64 | 11, 63 | ax-mp 5 | . . . . . . . . 9 |
65 | 19 | nnnn0d 9161 | . . . . . . . . 9 |
66 | bernneq3 10571 | . . . . . . . . 9 | |
67 | 64, 65, 66 | sylancr 411 | . . . . . . . 8 |
68 | 33, 30, 27 | ltdiv2d 9650 | . . . . . . . 8 |
69 | 67, 68 | mpbid 146 | . . . . . . 7 |
70 | 26, 32, 35, 62, 69 | lttrd 8018 | . . . . . 6 |
71 | 21, 25, 35 | ltsubadd2d 8435 | . . . . . 6 |
72 | 70, 71 | mpbid 146 | . . . . 5 |
73 | 21, 35 | readdcld 7922 | . . . . . 6 |
74 | 25 | adantr 274 | . . . . . . . 8 |
75 | 21 | adantr 274 | . . . . . . . 8 |
76 | 36 | adantr 274 | . . . . . . . . 9 |
77 | 37 | adantr 274 | . . . . . . . . 9 |
78 | 19 | adantr 274 | . . . . . . . . 9 |
79 | 23 | adantr 274 | . . . . . . . . 9 |
80 | simpr 109 | . . . . . . . . 9 | |
81 | 1, 76, 77, 78, 79, 80 | resqrexlemdecn 10948 | . . . . . . . 8 |
82 | 74, 75, 81 | ltled 8011 | . . . . . . 7 |
83 | fveq2 5483 | . . . . . . . . 9 | |
84 | 83 | eqcomd 2170 | . . . . . . . 8 |
85 | eqle 7984 | . . . . . . . 8 | |
86 | 25, 84, 85 | syl2an 287 | . . . . . . 7 |
87 | 23 | nnzd 9306 | . . . . . . . . 9 |
88 | zleloe 9232 | . . . . . . . . 9 | |
89 | 29, 87, 88 | syl2anc 409 | . . . . . . . 8 |
90 | 39, 89 | mpbid 146 | . . . . . . 7 |
91 | 82, 86, 90 | mpjaodan 788 | . . . . . 6 |
92 | 21, 34 | ltaddrpd 9660 | . . . . . 6 |
93 | 25, 21, 73, 91, 92 | lelttrd 8017 | . . . . 5 |
94 | 72, 93 | jca 304 | . . . 4 |
95 | 94 | ralrimiva 2537 | . . 3 |
96 | 95 | ralrimiva 2537 | . 2 |
97 | 7, 17, 96 | cvg1n 10922 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wral 2442 wrex 2443 wss 3114 csn 3573 class class class wbr 3979 cxp 4599 wf 5181 cfv 5185 (class class class)co 5839 cmpo 5841 cc 7745 cr 7746 cc0 7747 c1 7748 caddc 7750 cmul 7752 clt 7927 cle 7928 cmin 8063 # cap 8473 cdiv 8562 cn 8851 c2 8902 cn0 9108 cz 9185 cuz 9460 crp 9583 cseq 10374 cexp 10448 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4094 ax-sep 4097 ax-nul 4105 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-iinf 4562 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-0lt1 7853 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltirr 7859 ax-pre-ltwlin 7860 ax-pre-lttrn 7861 ax-pre-apti 7862 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 ax-pre-mulext 7865 ax-arch 7866 ax-caucvg 7867 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2726 df-sbc 2950 df-csb 3044 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-if 3519 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-iun 3865 df-br 3980 df-opab 4041 df-mpt 4042 df-tr 4078 df-id 4268 df-po 4271 df-iso 4272 df-iord 4341 df-on 4343 df-ilim 4344 df-suc 4346 df-iom 4565 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-rn 4612 df-res 4613 df-ima 4614 df-iota 5150 df-fun 5187 df-fn 5188 df-f 5189 df-f1 5190 df-fo 5191 df-f1o 5192 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-1st 6103 df-2nd 6104 df-recs 6267 df-frec 6353 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 df-reap 8467 df-ap 8474 df-div 8563 df-inn 8852 df-2 8910 df-3 8911 df-4 8912 df-n0 9109 df-z 9186 df-uz 9461 df-rp 9584 df-seqfrec 10375 df-exp 10449 |
This theorem is referenced by: resqrexlemex 10961 |
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