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Mirrors > Home > ILE Home > Th. List > ser3mono | Unicode version |
Description: The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
Ref | Expression |
---|---|
sermono.1 | |
sermono.2 | |
ser3mono.3 | |
sermono.4 |
Ref | Expression |
---|---|
ser3mono |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sermono.2 | . 2 | |
2 | eqid 2139 | . . . 4 | |
3 | sermono.1 | . . . . . 6 | |
4 | eluzel2 9334 | . . . . . 6 | |
5 | 3, 4 | syl 14 | . . . . 5 |
6 | 5 | adantr 274 | . . . 4 |
7 | ser3mono.3 | . . . . 5 | |
8 | 7 | adantlr 468 | . . . 4 |
9 | 2, 6, 8 | serfre 10251 | . . 3 |
10 | elfzuz 9805 | . . . 4 | |
11 | uztrn 9345 | . . . 4 | |
12 | 10, 3, 11 | syl2anr 288 | . . 3 |
13 | 9, 12 | ffvelrnd 5556 | . 2 |
14 | fveq2 5421 | . . . . . 6 | |
15 | 14 | breq2d 3941 | . . . . 5 |
16 | sermono.4 | . . . . . . 7 | |
17 | 16 | ralrimiva 2505 | . . . . . 6 |
18 | 17 | adantr 274 | . . . . 5 |
19 | simpr 109 | . . . . . . 7 | |
20 | 3 | adantr 274 | . . . . . . . . 9 |
21 | eluzelz 9338 | . . . . . . . . 9 | |
22 | 20, 21 | syl 14 | . . . . . . . 8 |
23 | 1 | adantr 274 | . . . . . . . . . 10 |
24 | eluzelz 9338 | . . . . . . . . . 10 | |
25 | 23, 24 | syl 14 | . . . . . . . . 9 |
26 | peano2zm 9095 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | elfzelz 9809 | . . . . . . . . 9 | |
29 | 28 | adantl 275 | . . . . . . . 8 |
30 | 1zzd 9084 | . . . . . . . 8 | |
31 | fzaddel 9842 | . . . . . . . 8 | |
32 | 22, 27, 29, 30, 31 | syl22anc 1217 | . . . . . . 7 |
33 | 19, 32 | mpbid 146 | . . . . . 6 |
34 | zcn 9062 | . . . . . . . . 9 | |
35 | ax-1cn 7716 | . . . . . . . . 9 | |
36 | npcan 7974 | . . . . . . . . 9 | |
37 | 34, 35, 36 | sylancl 409 | . . . . . . . 8 |
38 | 25, 37 | syl 14 | . . . . . . 7 |
39 | 38 | oveq2d 5790 | . . . . . 6 |
40 | 33, 39 | eleqtrd 2218 | . . . . 5 |
41 | 15, 18, 40 | rspcdva 2794 | . . . 4 |
42 | fzelp1 9857 | . . . . . . . 8 | |
43 | 42 | adantl 275 | . . . . . . 7 |
44 | 38 | oveq2d 5790 | . . . . . . 7 |
45 | 43, 44 | eleqtrd 2218 | . . . . . 6 |
46 | 45, 13 | syldan 280 | . . . . 5 |
47 | 14 | eleq1d 2208 | . . . . . 6 |
48 | 7 | ralrimiva 2505 | . . . . . . 7 |
49 | 48 | adantr 274 | . . . . . 6 |
50 | fzss1 9846 | . . . . . . . . 9 | |
51 | 20, 50 | syl 14 | . . . . . . . 8 |
52 | fzp1elp1 9858 | . . . . . . . . . 10 | |
53 | 52 | adantl 275 | . . . . . . . . 9 |
54 | 53, 44 | eleqtrd 2218 | . . . . . . . 8 |
55 | 51, 54 | sseldd 3098 | . . . . . . 7 |
56 | elfzuz 9805 | . . . . . . 7 | |
57 | 55, 56 | syl 14 | . . . . . 6 |
58 | 47, 49, 57 | rspcdva 2794 | . . . . 5 |
59 | 46, 58 | addge01d 8298 | . . . 4 |
60 | 41, 59 | mpbid 146 | . . 3 |
61 | 45, 12 | syldan 280 | . . . 4 |
62 | 7 | adantlr 468 | . . . 4 |
63 | readdcl 7749 | . . . . 5 | |
64 | 63 | adantl 275 | . . . 4 |
65 | 61, 62, 64 | seq3p1 10238 | . . 3 |
66 | 60, 65 | breqtrrd 3956 | . 2 |
67 | 1, 13, 66 | monoord 10252 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wss 3071 class class class wbr 3929 cfv 5123 (class class class)co 5774 cc 7621 cr 7622 cc0 7623 c1 7624 caddc 7626 cle 7804 cmin 7936 cz 9057 cuz 9329 cfz 9793 cseq 10221 |
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 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 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-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-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-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 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 df-seqfrec 10222 |
This theorem is referenced by: (None) |
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