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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 9767 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2180 | . . . . . 6 | |
5 | fveq2 5389 | . . . . . . 7 | |
6 | 5 | eqeq2d 2129 | . . . . . 6 |
7 | 4, 6 | imbi12d 233 | . . . . 5 |
8 | 7 | imbi2d 229 | . . . 4 |
9 | eleq1 2180 | . . . . . 6 | |
10 | fveq2 5389 | . . . . . . 7 | |
11 | 10 | eqeq2d 2129 | . . . . . 6 |
12 | 9, 11 | imbi12d 233 | . . . . 5 |
13 | 12 | imbi2d 229 | . . . 4 |
14 | eleq1 2180 | . . . . . 6 | |
15 | fveq2 5389 | . . . . . . 7 | |
16 | 15 | eqeq2d 2129 | . . . . . 6 |
17 | 14, 16 | imbi12d 233 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | eleq1 2180 | . . . . . 6 | |
20 | fveq2 5389 | . . . . . . 7 | |
21 | 20 | eqeq2d 2129 | . . . . . 6 |
22 | 19, 21 | imbi12d 233 | . . . . 5 |
23 | 22 | imbi2d 229 | . . . 4 |
24 | eqidd 2118 | . . . . 5 | |
25 | 24 | 2a1i 27 | . . . 4 |
26 | peano2fzr 9772 | . . . . . . . 8 | |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | expr 372 | . . . . . 6 |
29 | 28 | imim1d 75 | . . . . 5 |
30 | oveq1 5749 | . . . . . 6 | |
31 | fveqeq2 5398 | . . . . . . . . . 10 | |
32 | seqid2.5 | . . . . . . . . . . . 12 | |
33 | 32 | ralrimiva 2482 | . . . . . . . . . . 11 |
34 | 33 | adantr 274 | . . . . . . . . . 10 |
35 | eluzp1p1 9307 | . . . . . . . . . . . 12 | |
36 | 35 | ad2antrl 481 | . . . . . . . . . . 11 |
37 | elfzuz3 9758 | . . . . . . . . . . . 12 | |
38 | 37 | ad2antll 482 | . . . . . . . . . . 11 |
39 | elfzuzb 9755 | . . . . . . . . . . 11 | |
40 | 36, 38, 39 | sylanbrc 413 | . . . . . . . . . 10 |
41 | 31, 34, 40 | rspcdva 2768 | . . . . . . . . 9 |
42 | 41 | oveq2d 5758 | . . . . . . . 8 |
43 | oveq1 5749 | . . . . . . . . . . 11 | |
44 | id 19 | . . . . . . . . . . 11 | |
45 | 43, 44 | eqeq12d 2132 | . . . . . . . . . 10 |
46 | seqid2.1 | . . . . . . . . . . 11 | |
47 | 46 | ralrimiva 2482 | . . . . . . . . . 10 |
48 | seqid2.4 | . . . . . . . . . 10 | |
49 | 45, 47, 48 | rspcdva 2768 | . . . . . . . . 9 |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | 42, 50 | eqtr2d 2151 | . . . . . . 7 |
52 | simprl 505 | . . . . . . . . 9 | |
53 | seqid2.2 | . . . . . . . . . 10 | |
54 | 53 | adantr 274 | . . . . . . . . 9 |
55 | uztrn 9298 | . . . . . . . . 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 10190 | . . . . . . 7 |
62 | 51, 61 | eqeq12d 2132 | . . . . . 6 |
63 | 30, 62 | syl5ibr 155 | . . . . 5 |
64 | 29, 63 | animpimp2impd 533 | . . . 4 |
65 | 8, 13, 18, 23, 25, 64 | uzind4 9339 | . . 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 1316 wcel 1465 wral 2393 cfv 5093 (class class class)co 5742 c1 7589 caddc 7591 cz 9012 cuz 9282 cfz 9745 cseq 10173 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-fz 9746 df-seqfrec 10174 |
This theorem is referenced by: seq3coll 10540 fsum3cvg 11101 |
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