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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 9988 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2233 | . . . . . 6 | |
5 | fveq2 5496 | . . . . . . 7 | |
6 | fvoveq1 5876 | . . . . . . 7 | |
7 | 5, 6 | eqeq12d 2185 | . . . . . 6 |
8 | 4, 7 | imbi12d 233 | . . . . 5 |
9 | 8 | imbi2d 229 | . . . 4 |
10 | eleq1 2233 | . . . . . 6 | |
11 | fveq2 5496 | . . . . . . 7 | |
12 | fvoveq1 5876 | . . . . . . 7 | |
13 | 11, 12 | eqeq12d 2185 | . . . . . 6 |
14 | 10, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | imbi2d 229 | . . . 4 |
16 | eleq1 2233 | . . . . . 6 | |
17 | fveq2 5496 | . . . . . . 7 | |
18 | fvoveq1 5876 | . . . . . . 7 | |
19 | 17, 18 | eqeq12d 2185 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 20 | imbi2d 229 | . . . 4 |
22 | eleq1 2233 | . . . . . 6 | |
23 | fveq2 5496 | . . . . . . 7 | |
24 | fvoveq1 5876 | . . . . . . 7 | |
25 | 23, 24 | eqeq12d 2185 | . . . . . 6 |
26 | 22, 25 | imbi12d 233 | . . . . 5 |
27 | 26 | imbi2d 229 | . . . 4 |
28 | fveq2 5496 | . . . . . . . 8 | |
29 | fvoveq1 5876 | . . . . . . . 8 | |
30 | 28, 29 | eqeq12d 2185 | . . . . . . 7 |
31 | seq3shft2.3 | . . . . . . . 8 | |
32 | 31 | ralrimiva 2543 | . . . . . . 7 |
33 | eluzfz1 9987 | . . . . . . . 8 | |
34 | 1, 33 | syl 14 | . . . . . . 7 |
35 | 30, 32, 34 | rspcdva 2839 | . . . . . 6 |
36 | eluzel2 9492 | . . . . . . . 8 | |
37 | 1, 36 | syl 14 | . . . . . . 7 |
38 | seq3shft2.f | . . . . . . 7 | |
39 | seq3shft2.pl | . . . . . . 7 | |
40 | 37, 38, 39 | seq3-1 10416 | . . . . . 6 |
41 | seq3shft2.2 | . . . . . . . 8 | |
42 | 37, 41 | zaddcld 9338 | . . . . . . 7 |
43 | seq3shft2.g | . . . . . . 7 | |
44 | 42, 43, 39 | seq3-1 10416 | . . . . . 6 |
45 | 35, 40, 44 | 3eqtr4d 2213 | . . . . 5 |
46 | 45 | a1i13 24 | . . . 4 |
47 | peano2fzr 9993 | . . . . . . . 8 | |
48 | 47 | adantl 275 | . . . . . . 7 |
49 | 48 | expr 373 | . . . . . 6 |
50 | 49 | imim1d 75 | . . . . 5 |
51 | oveq1 5860 | . . . . . 6 | |
52 | simprl 526 | . . . . . . . 8 | |
53 | 38 | adantlr 474 | . . . . . . . 8 |
54 | 39 | adantlr 474 | . . . . . . . 8 |
55 | 52, 53, 54 | seq3p1 10418 | . . . . . . 7 |
56 | 41 | adantr 274 | . . . . . . . . . 10 |
57 | eluzadd 9515 | . . . . . . . . . 10 | |
58 | 52, 56, 57 | syl2anc 409 | . . . . . . . . 9 |
59 | 43 | adantlr 474 | . . . . . . . . 9 |
60 | 58, 59, 54 | seq3p1 10418 | . . . . . . . 8 |
61 | eluzelz 9496 | . . . . . . . . . . . 12 | |
62 | 52, 61 | syl 14 | . . . . . . . . . . 11 |
63 | 62 | zcnd 9335 | . . . . . . . . . 10 |
64 | 1cnd 7936 | . . . . . . . . . 10 | |
65 | 56 | zcnd 9335 | . . . . . . . . . 10 |
66 | 63, 64, 65 | add32d 8087 | . . . . . . . . 9 |
67 | 66 | fveq2d 5500 | . . . . . . . 8 |
68 | fveq2 5496 | . . . . . . . . . . . 12 | |
69 | fvoveq1 5876 | . . . . . . . . . . . 12 | |
70 | 68, 69 | eqeq12d 2185 | . . . . . . . . . . 11 |
71 | 32 | adantr 274 | . . . . . . . . . . 11 |
72 | simprr 527 | . . . . . . . . . . 11 | |
73 | 70, 71, 72 | rspcdva 2839 | . . . . . . . . . 10 |
74 | 66 | fveq2d 5500 | . . . . . . . . . 10 |
75 | 73, 74 | eqtrd 2203 | . . . . . . . . 9 |
76 | 75 | oveq2d 5869 | . . . . . . . 8 |
77 | 60, 67, 76 | 3eqtr4d 2213 | . . . . . . 7 |
78 | 55, 77 | eqeq12d 2185 | . . . . . 6 |
79 | 51, 78 | syl5ibr 155 | . . . . 5 |
80 | 50, 79 | animpimp2impd 554 | . . . 4 |
81 | 9, 15, 21, 27, 46, 80 | uzind4 9547 | . . 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 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: seq3f1olemqsumkj 10454 seq3shft 10802 |
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