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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 9988 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2233 | . . . . . 6 | |
5 | fveq2 5496 | . . . . . . 7 | |
6 | 5 | eqeq2d 2182 | . . . . . 6 |
7 | 4, 6 | imbi12d 233 | . . . . 5 |
8 | 7 | imbi2d 229 | . . . 4 |
9 | eleq1 2233 | . . . . . 6 | |
10 | fveq2 5496 | . . . . . . 7 | |
11 | 10 | eqeq2d 2182 | . . . . . 6 |
12 | 9, 11 | imbi12d 233 | . . . . 5 |
13 | 12 | imbi2d 229 | . . . 4 |
14 | eleq1 2233 | . . . . . 6 | |
15 | fveq2 5496 | . . . . . . 7 | |
16 | 15 | eqeq2d 2182 | . . . . . 6 |
17 | 14, 16 | imbi12d 233 | . . . . 5 |
18 | 17 | imbi2d 229 | . . . 4 |
19 | eleq1 2233 | . . . . . 6 | |
20 | fveq2 5496 | . . . . . . 7 | |
21 | 20 | eqeq2d 2182 | . . . . . 6 |
22 | 19, 21 | imbi12d 233 | . . . . 5 |
23 | 22 | imbi2d 229 | . . . 4 |
24 | eqidd 2171 | . . . . 5 | |
25 | 24 | 2a1i 27 | . . . 4 |
26 | peano2fzr 9993 | . . . . . . . 8 | |
27 | 26 | adantl 275 | . . . . . . 7 |
28 | 27 | expr 373 | . . . . . 6 |
29 | 28 | imim1d 75 | . . . . 5 |
30 | oveq1 5860 | . . . . . 6 | |
31 | fveqeq2 5505 | . . . . . . . . . 10 | |
32 | seqid2.5 | . . . . . . . . . . . 12 | |
33 | 32 | ralrimiva 2543 | . . . . . . . . . . 11 |
34 | 33 | adantr 274 | . . . . . . . . . 10 |
35 | eluzp1p1 9512 | . . . . . . . . . . . 12 | |
36 | 35 | ad2antrl 487 | . . . . . . . . . . 11 |
37 | elfzuz3 9978 | . . . . . . . . . . . 12 | |
38 | 37 | ad2antll 488 | . . . . . . . . . . 11 |
39 | elfzuzb 9975 | . . . . . . . . . . 11 | |
40 | 36, 38, 39 | sylanbrc 415 | . . . . . . . . . 10 |
41 | 31, 34, 40 | rspcdva 2839 | . . . . . . . . 9 |
42 | 41 | oveq2d 5869 | . . . . . . . 8 |
43 | oveq1 5860 | . . . . . . . . . . 11 | |
44 | id 19 | . . . . . . . . . . 11 | |
45 | 43, 44 | eqeq12d 2185 | . . . . . . . . . 10 |
46 | seqid2.1 | . . . . . . . . . . 11 | |
47 | 46 | ralrimiva 2543 | . . . . . . . . . 10 |
48 | seqid2.4 | . . . . . . . . . 10 | |
49 | 45, 47, 48 | rspcdva 2839 | . . . . . . . . 9 |
50 | 49 | adantr 274 | . . . . . . . 8 |
51 | 42, 50 | eqtr2d 2204 | . . . . . . 7 |
52 | simprl 526 | . . . . . . . . 9 | |
53 | seqid2.2 | . . . . . . . . . 10 | |
54 | 53 | adantr 274 | . . . . . . . . 9 |
55 | uztrn 9503 | . . . . . . . . 9 | |
56 | 52, 54, 55 | syl2anc 409 | . . . . . . . 8 |
57 | seq3id2.f | . . . . . . . . 9 | |
58 | 57 | adantlr 474 | . . . . . . . 8 |
59 | seq3id2.cl | . . . . . . . . 9 | |
60 | 59 | adantlr 474 | . . . . . . . 8 |
61 | 56, 58, 60 | seq3p1 10418 | . . . . . . 7 |
62 | 51, 61 | eqeq12d 2185 | . . . . . 6 |
63 | 30, 62 | syl5ibr 155 | . . . . 5 |
64 | 29, 63 | animpimp2impd 554 | . . . 4 |
65 | 8, 13, 18, 23, 25, 64 | uzind4 9547 | . . 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 1348 wcel 2141 wral 2448 cfv 5198 (class class class)co 5853 c1 7775 caddc 7777 cz 9212 cuz 9487 cfz 9965 cseq 10401 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-seqfrec 10402 |
This theorem is referenced by: seq3coll 10777 fsum3cvg 11341 fproddccvg 11535 lgsdilem2 13731 |
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