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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 2157 | . . . 4 | |
2 | seq3shft.m | . . . 4 | |
3 | seq3shft.ex | . . . . . . 7 | |
4 | 3 | adantr 274 | . . . . . 6 |
5 | seq3shft.n | . . . . . . . 8 | |
6 | 5 | zcnd 9287 | . . . . . . 7 |
7 | 6 | adantr 274 | . . . . . 6 |
8 | eluzelz 9448 | . . . . . . . 8 | |
9 | 8 | adantl 275 | . . . . . . 7 |
10 | 9 | zcnd 9287 | . . . . . 6 |
11 | shftvalg 10736 | . . . . . 6 | |
12 | 4, 7, 10, 11 | syl3anc 1220 | . . . . 5 |
13 | fveq2 5468 | . . . . . . 7 | |
14 | 13 | eleq1d 2226 | . . . . . 6 |
15 | seq3shft.fn | . . . . . . . . 9 | |
16 | 15 | ralrimiva 2530 | . . . . . . . 8 |
17 | fveq2 5468 | . . . . . . . . . 10 | |
18 | 17 | eleq1d 2226 | . . . . . . . . 9 |
19 | 18 | cbvralv 2680 | . . . . . . . 8 |
20 | 16, 19 | sylib 121 | . . . . . . 7 |
21 | 20 | adantr 274 | . . . . . 6 |
22 | 2, 5 | zsubcld 9291 | . . . . . . . 8 |
23 | 22 | adantr 274 | . . . . . . 7 |
24 | 5 | adantr 274 | . . . . . . . 8 |
25 | 9, 24 | zsubcld 9291 | . . . . . . 7 |
26 | 2 | zred 9286 | . . . . . . . . 9 |
27 | 26 | adantr 274 | . . . . . . . 8 |
28 | 9 | zred 9286 | . . . . . . . 8 |
29 | 24 | zred 9286 | . . . . . . . 8 |
30 | eluzle 9451 | . . . . . . . . 9 | |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | 27, 28, 29, 31 | lesub1dd 8436 | . . . . . . 7 |
33 | eluz2 9445 | . . . . . . 7 | |
34 | 23, 25, 32, 33 | syl3anbrc 1166 | . . . . . 6 |
35 | 14, 21, 34 | rspcdva 2821 | . . . . 5 |
36 | 12, 35 | eqeltrd 2234 | . . . 4 |
37 | seq3shft.pl | . . . 4 | |
38 | 1, 2, 36, 37 | seqf 10360 | . . 3 |
39 | 38 | ffnd 5320 | . 2 |
40 | eqid 2157 | . . . . . 6 | |
41 | 40, 22, 15, 37 | seqf 10360 | . . . . 5 |
42 | 41 | ffnd 5320 | . . . 4 |
43 | seqex 10346 | . . . . 5 | |
44 | 43 | shftfn 10724 | . . . 4 |
45 | 42, 6, 44 | syl2anc 409 | . . 3 |
46 | shftuz 10717 | . . . . . 6 | |
47 | 5, 22, 46 | syl2anc 409 | . . . . 5 |
48 | 2 | zcnd 9287 | . . . . . . 7 |
49 | 48, 6 | npcand 8190 | . . . . . 6 |
50 | 49 | fveq2d 5472 | . . . . 5 |
51 | 47, 50 | eqtrd 2190 | . . . 4 |
52 | 51 | fneq2d 5261 | . . 3 |
53 | 45, 52 | mpbid 146 | . 2 |
54 | 48, 6 | negsubd 8192 | . . . . . 6 |
55 | 54 | adantr 274 | . . . . 5 |
56 | 55 | seqeq1d 10350 | . . . 4 |
57 | eluzelcn 9450 | . . . . . 6 | |
58 | 57 | adantl 275 | . . . . 5 |
59 | 6 | adantr 274 | . . . . 5 |
60 | 58, 59 | negsubd 8192 | . . . 4 |
61 | 56, 60 | fveq12d 5475 | . . 3 |
62 | simpr 109 | . . . 4 | |
63 | 5 | adantr 274 | . . . . 5 |
64 | 63 | znegcld 9288 | . . . 4 |
65 | 3 | ad2antrr 480 | . . . . . 6 |
66 | 59 | adantr 274 | . . . . . 6 |
67 | elfzelz 9928 | . . . . . . . 8 | |
68 | 67 | adantl 275 | . . . . . . 7 |
69 | 68 | zcnd 9287 | . . . . . 6 |
70 | shftvalg 10736 | . . . . . 6 | |
71 | 65, 66, 69, 70 | syl3anc 1220 | . . . . 5 |
72 | 69, 66 | negsubd 8192 | . . . . . 6 |
73 | 72 | fveq2d 5472 | . . . . 5 |
74 | 71, 73 | eqtr4d 2193 | . . . 4 |
75 | 36 | adantlr 469 | . . . 4 |
76 | simpll 519 | . . . . 5 | |
77 | simpr 109 | . . . . . 6 | |
78 | 54 | fveq2d 5472 | . . . . . . . 8 |
79 | 78 | eleq2d 2227 | . . . . . . 7 |
80 | 79 | ad2antrr 480 | . . . . . 6 |
81 | 77, 80 | mpbid 146 | . . . . 5 |
82 | 76, 81, 15 | syl2anc 409 | . . . 4 |
83 | 37 | adantlr 469 | . . . 4 |
84 | 62, 64, 74, 75, 82, 83 | seq3shft2 10372 | . . 3 |
85 | shftvalg 10736 | . . . 4 | |
86 | 43, 59, 58, 85 | mp3an2i 1324 | . . 3 |
87 | 61, 84, 86 | 3eqtr4d 2200 | . 2 |
88 | 39, 53, 87 | eqfnfvd 5568 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wral 2435 crab 2439 cvv 2712 class class class wbr 3965 wfn 5165 cfv 5170 (class class class)co 5824 cc 7730 cr 7731 caddc 7735 cle 7913 cmin 8046 cneg 8047 cz 9167 cuz 9439 cfz 9912 cseq 10344 cshi 10714 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-fz 9913 df-seqfrec 10345 df-shft 10715 |
This theorem is referenced by: iser3shft 11243 eftlub 11587 |
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