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Mirrors > Home > ILE Home > Th. List > seq3id2 | Unicode version |
Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for ) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.) |
Ref | Expression |
---|---|
seqid2.1 | |
seqid2.2 | |
seqid2.3 | |
seqid2.4 | |
seqid2.5 | |
seq3id2.f | |
seq3id2.cl |
Ref | Expression |
---|---|
seq3id2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seqid2.3 | . . 3 | |
2 | eluzfz2 9812 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2202 | . . . . . 6 | |
5 | fveq2 5421 | . . . . . . 7 | |
6 | 5 | eqeq2d 2151 | . . . . . 6 |
7 | 4, 6 | imbi12d 233 | . . . . 5 |
8 | 7 | imbi2d 229 | . . . 4 |
9 | eleq1 2202 | . . . . . 6 | |
10 | fveq2 5421 | . . . . . . 7 | |
11 | 10 | eqeq2d 2151 | . . . . . 6 |
12 | 9, 11 | imbi12d 233 | . . . . 5 |
13 | 12 | imbi2d 229 | . . . 4 |
14 | eleq1 2202 | . . . . . 6 | |
15 | fveq2 5421 | . . . . . . 7 | |
16 | 15 | eqeq2d 2151 | . . . . . 6 |
17 | 14, 16 | imbi12d 233 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | eleq1 2202 | . . . . . 6 | |
20 | fveq2 5421 | . . . . . . 7 | |
21 | 20 | eqeq2d 2151 | . . . . . 6 |
22 | 19, 21 | imbi12d 233 | . . . . 5 |
23 | 22 | imbi2d 229 | . . . 4 |
24 | eqidd 2140 | . . . . 5 | |
25 | 24 | 2a1i 27 | . . . 4 |
26 | peano2fzr 9817 | . . . . . . . 8 | |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | expr 372 | . . . . . 6 |
29 | 28 | imim1d 75 | . . . . 5 |
30 | oveq1 5781 | . . . . . 6 | |
31 | fveqeq2 5430 | . . . . . . . . . 10 | |
32 | seqid2.5 | . . . . . . . . . . . 12 | |
33 | 32 | ralrimiva 2505 | . . . . . . . . . . 11 |
34 | 33 | adantr 274 | . . . . . . . . . 10 |
35 | eluzp1p1 9351 | . . . . . . . . . . . 12 | |
36 | 35 | ad2antrl 481 | . . . . . . . . . . 11 |
37 | elfzuz3 9803 | . . . . . . . . . . . 12 | |
38 | 37 | ad2antll 482 | . . . . . . . . . . 11 |
39 | elfzuzb 9800 | . . . . . . . . . . 11 | |
40 | 36, 38, 39 | sylanbrc 413 | . . . . . . . . . 10 |
41 | 31, 34, 40 | rspcdva 2794 | . . . . . . . . 9 |
42 | 41 | oveq2d 5790 | . . . . . . . 8 |
43 | oveq1 5781 | . . . . . . . . . . 11 | |
44 | id 19 | . . . . . . . . . . 11 | |
45 | 43, 44 | eqeq12d 2154 | . . . . . . . . . 10 |
46 | seqid2.1 | . . . . . . . . . . 11 | |
47 | 46 | ralrimiva 2505 | . . . . . . . . . 10 |
48 | seqid2.4 | . . . . . . . . . 10 | |
49 | 45, 47, 48 | rspcdva 2794 | . . . . . . . . 9 |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | 42, 50 | eqtr2d 2173 | . . . . . . 7 |
52 | simprl 520 | . . . . . . . . 9 | |
53 | seqid2.2 | . . . . . . . . . 10 | |
54 | 53 | adantr 274 | . . . . . . . . 9 |
55 | uztrn 9342 | . . . . . . . . 9 | |
56 | 52, 54, 55 | syl2anc 408 | . . . . . . . 8 |
57 | seq3id2.f | . . . . . . . . 9 | |
58 | 57 | adantlr 468 | . . . . . . . 8 |
59 | seq3id2.cl | . . . . . . . . 9 | |
60 | 59 | adantlr 468 | . . . . . . . 8 |
61 | 56, 58, 60 | seq3p1 10235 | . . . . . . 7 |
62 | 51, 61 | eqeq12d 2154 | . . . . . 6 |
63 | 30, 62 | syl5ibr 155 | . . . . 5 |
64 | 29, 63 | animpimp2impd 548 | . . . 4 |
65 | 8, 13, 18, 23, 25, 64 | uzind4 9383 | . . 3 |
66 | 1, 65 | mpcom 36 | . 2 |
67 | 3, 66 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 cfv 5123 (class class class)co 5774 c1 7621 caddc 7623 cz 9054 cuz 9326 cfz 9790 cseq 10218 |
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-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-seqfrec 10219 |
This theorem is referenced by: seq3coll 10585 fsum3cvg 11147 fproddccvg 11341 |
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