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Mirrors > Home > ILE Home > Th. List > seq3shft2 | Unicode version |
Description: Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
Ref | Expression |
---|---|
seq3shft2.1 | |
seq3shft2.2 | |
seq3shft2.3 | |
seq3shft2.f | |
seq3shft2.g | |
seq3shft2.pl |
Ref | Expression |
---|---|
seq3shft2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq3shft2.1 | . . 3 | |
2 | eluzfz2 9935 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2220 | . . . . . 6 | |
5 | fveq2 5469 | . . . . . . 7 | |
6 | fvoveq1 5848 | . . . . . . 7 | |
7 | 5, 6 | eqeq12d 2172 | . . . . . 6 |
8 | 4, 7 | imbi12d 233 | . . . . 5 |
9 | 8 | imbi2d 229 | . . . 4 |
10 | eleq1 2220 | . . . . . 6 | |
11 | fveq2 5469 | . . . . . . 7 | |
12 | fvoveq1 5848 | . . . . . . 7 | |
13 | 11, 12 | eqeq12d 2172 | . . . . . 6 |
14 | 10, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | imbi2d 229 | . . . 4 |
16 | eleq1 2220 | . . . . . 6 | |
17 | fveq2 5469 | . . . . . . 7 | |
18 | fvoveq1 5848 | . . . . . . 7 | |
19 | 17, 18 | eqeq12d 2172 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 20 | imbi2d 229 | . . . 4 |
22 | eleq1 2220 | . . . . . 6 | |
23 | fveq2 5469 | . . . . . . 7 | |
24 | fvoveq1 5848 | . . . . . . 7 | |
25 | 23, 24 | eqeq12d 2172 | . . . . . 6 |
26 | 22, 25 | imbi12d 233 | . . . . 5 |
27 | 26 | imbi2d 229 | . . . 4 |
28 | fveq2 5469 | . . . . . . . 8 | |
29 | fvoveq1 5848 | . . . . . . . 8 | |
30 | 28, 29 | eqeq12d 2172 | . . . . . . 7 |
31 | seq3shft2.3 | . . . . . . . 8 | |
32 | 31 | ralrimiva 2530 | . . . . . . 7 |
33 | eluzfz1 9934 | . . . . . . . 8 | |
34 | 1, 33 | syl 14 | . . . . . . 7 |
35 | 30, 32, 34 | rspcdva 2821 | . . . . . 6 |
36 | eluzel2 9445 | . . . . . . . 8 | |
37 | 1, 36 | syl 14 | . . . . . . 7 |
38 | seq3shft2.f | . . . . . . 7 | |
39 | seq3shft2.pl | . . . . . . 7 | |
40 | 37, 38, 39 | seq3-1 10363 | . . . . . 6 |
41 | seq3shft2.2 | . . . . . . . 8 | |
42 | 37, 41 | zaddcld 9291 | . . . . . . 7 |
43 | seq3shft2.g | . . . . . . 7 | |
44 | 42, 43, 39 | seq3-1 10363 | . . . . . 6 |
45 | 35, 40, 44 | 3eqtr4d 2200 | . . . . 5 |
46 | 45 | a1i13 24 | . . . 4 |
47 | peano2fzr 9940 | . . . . . . . 8 | |
48 | 47 | adantl 275 | . . . . . . 7 |
49 | 48 | expr 373 | . . . . . 6 |
50 | 49 | imim1d 75 | . . . . 5 |
51 | oveq1 5832 | . . . . . 6 | |
52 | simprl 521 | . . . . . . . 8 | |
53 | 38 | adantlr 469 | . . . . . . . 8 |
54 | 39 | adantlr 469 | . . . . . . . 8 |
55 | 52, 53, 54 | seq3p1 10365 | . . . . . . 7 |
56 | 41 | adantr 274 | . . . . . . . . . 10 |
57 | eluzadd 9468 | . . . . . . . . . 10 | |
58 | 52, 56, 57 | syl2anc 409 | . . . . . . . . 9 |
59 | 43 | adantlr 469 | . . . . . . . . 9 |
60 | 58, 59, 54 | seq3p1 10365 | . . . . . . . 8 |
61 | eluzelz 9449 | . . . . . . . . . . . 12 | |
62 | 52, 61 | syl 14 | . . . . . . . . . . 11 |
63 | 62 | zcnd 9288 | . . . . . . . . . 10 |
64 | 1cnd 7895 | . . . . . . . . . 10 | |
65 | 56 | zcnd 9288 | . . . . . . . . . 10 |
66 | 63, 64, 65 | add32d 8044 | . . . . . . . . 9 |
67 | 66 | fveq2d 5473 | . . . . . . . 8 |
68 | fveq2 5469 | . . . . . . . . . . . 12 | |
69 | fvoveq1 5848 | . . . . . . . . . . . 12 | |
70 | 68, 69 | eqeq12d 2172 | . . . . . . . . . . 11 |
71 | 32 | adantr 274 | . . . . . . . . . . 11 |
72 | simprr 522 | . . . . . . . . . . 11 | |
73 | 70, 71, 72 | rspcdva 2821 | . . . . . . . . . 10 |
74 | 66 | fveq2d 5473 | . . . . . . . . . 10 |
75 | 73, 74 | eqtrd 2190 | . . . . . . . . 9 |
76 | 75 | oveq2d 5841 | . . . . . . . 8 |
77 | 60, 67, 76 | 3eqtr4d 2200 | . . . . . . 7 |
78 | 55, 77 | eqeq12d 2172 | . . . . . 6 |
79 | 51, 78 | syl5ibr 155 | . . . . 5 |
80 | 50, 79 | animpimp2impd 549 | . . . 4 |
81 | 9, 15, 21, 27, 46, 80 | uzind4 9500 | . . 3 |
82 | 1, 81 | mpcom 36 | . 2 |
83 | 3, 82 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 cfv 5171 (class class class)co 5825 c1 7734 caddc 7736 cz 9168 cuz 9440 cfz 9913 cseq 10348 |
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 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
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 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-fz 9914 df-seqfrec 10349 |
This theorem is referenced by: seq3f1olemqsumkj 10401 seq3shft 10742 |
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