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Mirrors > Home > ILE Home > Th. List > seq3shft | Unicode version |
Description: Shifting the index set of a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 17-Oct-2022.) |
Ref | Expression |
---|---|
seq3shft.ex | |
seq3shft.m | |
seq3shft.n | |
seq3shft.fn | |
seq3shft.pl |
Ref | Expression |
---|---|
seq3shft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . . 4 | |
2 | seq3shft.m | . . . 4 | |
3 | seq3shft.ex | . . . . . . 7 | |
4 | 3 | adantr 274 | . . . . . 6 |
5 | seq3shft.n | . . . . . . . 8 | |
6 | 5 | zcnd 9167 | . . . . . . 7 |
7 | 6 | adantr 274 | . . . . . 6 |
8 | eluzelz 9328 | . . . . . . . 8 | |
9 | 8 | adantl 275 | . . . . . . 7 |
10 | 9 | zcnd 9167 | . . . . . 6 |
11 | shftvalg 10601 | . . . . . 6 | |
12 | 4, 7, 10, 11 | syl3anc 1216 | . . . . 5 |
13 | fveq2 5414 | . . . . . . 7 | |
14 | 13 | eleq1d 2206 | . . . . . 6 |
15 | seq3shft.fn | . . . . . . . . 9 | |
16 | 15 | ralrimiva 2503 | . . . . . . . 8 |
17 | fveq2 5414 | . . . . . . . . . 10 | |
18 | 17 | eleq1d 2206 | . . . . . . . . 9 |
19 | 18 | cbvralv 2652 | . . . . . . . 8 |
20 | 16, 19 | sylib 121 | . . . . . . 7 |
21 | 20 | adantr 274 | . . . . . 6 |
22 | 2, 5 | zsubcld 9171 | . . . . . . . 8 |
23 | 22 | adantr 274 | . . . . . . 7 |
24 | 5 | adantr 274 | . . . . . . . 8 |
25 | 9, 24 | zsubcld 9171 | . . . . . . 7 |
26 | 2 | zred 9166 | . . . . . . . . 9 |
27 | 26 | adantr 274 | . . . . . . . 8 |
28 | 9 | zred 9166 | . . . . . . . 8 |
29 | 24 | zred 9166 | . . . . . . . 8 |
30 | eluzle 9331 | . . . . . . . . 9 | |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | 27, 28, 29, 31 | lesub1dd 8316 | . . . . . . 7 |
33 | eluz2 9325 | . . . . . . 7 | |
34 | 23, 25, 32, 33 | syl3anbrc 1165 | . . . . . 6 |
35 | 14, 21, 34 | rspcdva 2789 | . . . . 5 |
36 | 12, 35 | eqeltrd 2214 | . . . 4 |
37 | seq3shft.pl | . . . 4 | |
38 | 1, 2, 36, 37 | seqf 10227 | . . 3 |
39 | 38 | ffnd 5268 | . 2 |
40 | eqid 2137 | . . . . . 6 | |
41 | 40, 22, 15, 37 | seqf 10227 | . . . . 5 |
42 | 41 | ffnd 5268 | . . . 4 |
43 | seqex 10213 | . . . . 5 | |
44 | 43 | shftfn 10589 | . . . 4 |
45 | 42, 6, 44 | syl2anc 408 | . . 3 |
46 | shftuz 10582 | . . . . . 6 | |
47 | 5, 22, 46 | syl2anc 408 | . . . . 5 |
48 | 2 | zcnd 9167 | . . . . . . 7 |
49 | 48, 6 | npcand 8070 | . . . . . 6 |
50 | 49 | fveq2d 5418 | . . . . 5 |
51 | 47, 50 | eqtrd 2170 | . . . 4 |
52 | 51 | fneq2d 5209 | . . 3 |
53 | 45, 52 | mpbid 146 | . 2 |
54 | 48, 6 | negsubd 8072 | . . . . . 6 |
55 | 54 | adantr 274 | . . . . 5 |
56 | 55 | seqeq1d 10217 | . . . 4 |
57 | eluzelcn 9330 | . . . . . 6 | |
58 | 57 | adantl 275 | . . . . 5 |
59 | 6 | adantr 274 | . . . . 5 |
60 | 58, 59 | negsubd 8072 | . . . 4 |
61 | 56, 60 | fveq12d 5421 | . . 3 |
62 | simpr 109 | . . . 4 | |
63 | 5 | adantr 274 | . . . . 5 |
64 | 63 | znegcld 9168 | . . . 4 |
65 | 3 | ad2antrr 479 | . . . . . 6 |
66 | 59 | adantr 274 | . . . . . 6 |
67 | elfzelz 9799 | . . . . . . . 8 | |
68 | 67 | adantl 275 | . . . . . . 7 |
69 | 68 | zcnd 9167 | . . . . . 6 |
70 | shftvalg 10601 | . . . . . 6 | |
71 | 65, 66, 69, 70 | syl3anc 1216 | . . . . 5 |
72 | 69, 66 | negsubd 8072 | . . . . . 6 |
73 | 72 | fveq2d 5418 | . . . . 5 |
74 | 71, 73 | eqtr4d 2173 | . . . 4 |
75 | 36 | adantlr 468 | . . . 4 |
76 | simpll 518 | . . . . 5 | |
77 | simpr 109 | . . . . . 6 | |
78 | 54 | fveq2d 5418 | . . . . . . . 8 |
79 | 78 | eleq2d 2207 | . . . . . . 7 |
80 | 79 | ad2antrr 479 | . . . . . 6 |
81 | 77, 80 | mpbid 146 | . . . . 5 |
82 | 76, 81, 15 | syl2anc 408 | . . . 4 |
83 | 37 | adantlr 468 | . . . 4 |
84 | 62, 64, 74, 75, 82, 83 | seq3shft2 10239 | . . 3 |
85 | shftvalg 10601 | . . . 4 | |
86 | 43, 59, 58, 85 | mp3an2i 1320 | . . 3 |
87 | 61, 84, 86 | 3eqtr4d 2180 | . 2 |
88 | 39, 53, 87 | eqfnfvd 5514 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 crab 2418 cvv 2681 class class class wbr 3924 wfn 5113 cfv 5118 (class class class)co 5767 cc 7611 cr 7612 caddc 7616 cle 7794 cmin 7926 cneg 7927 cz 9047 cuz 9319 cfz 9783 cseq 10211 cshi 10579 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-seqfrec 10212 df-shft 10580 |
This theorem is referenced by: iser3shft 11108 eftlub 11385 |
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