Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9812 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | eleq1 2202 | . . . . . 6 | |
5 | fveq2 5421 | . . . . . . 7 | |
6 | fvoveq1 5797 | . . . . . . 7 | |
7 | 5, 6 | eqeq12d 2154 | . . . . . 6 |
8 | 4, 7 | imbi12d 233 | . . . . 5 |
9 | 8 | imbi2d 229 | . . . 4 |
10 | eleq1 2202 | . . . . . 6 | |
11 | fveq2 5421 | . . . . . . 7 | |
12 | fvoveq1 5797 | . . . . . . 7 | |
13 | 11, 12 | eqeq12d 2154 | . . . . . 6 |
14 | 10, 13 | imbi12d 233 | . . . . 5 |
15 | 14 | imbi2d 229 | . . . 4 |
16 | eleq1 2202 | . . . . . 6 | |
17 | fveq2 5421 | . . . . . . 7 | |
18 | fvoveq1 5797 | . . . . . . 7 | |
19 | 17, 18 | eqeq12d 2154 | . . . . . 6 |
20 | 16, 19 | imbi12d 233 | . . . . 5 |
21 | 20 | imbi2d 229 | . . . 4 |
22 | eleq1 2202 | . . . . . 6 | |
23 | fveq2 5421 | . . . . . . 7 | |
24 | fvoveq1 5797 | . . . . . . 7 | |
25 | 23, 24 | eqeq12d 2154 | . . . . . 6 |
26 | 22, 25 | imbi12d 233 | . . . . 5 |
27 | 26 | imbi2d 229 | . . . 4 |
28 | fveq2 5421 | . . . . . . . 8 | |
29 | fvoveq1 5797 | . . . . . . . 8 | |
30 | 28, 29 | eqeq12d 2154 | . . . . . . 7 |
31 | seq3shft2.3 | . . . . . . . 8 | |
32 | 31 | ralrimiva 2505 | . . . . . . 7 |
33 | eluzfz1 9811 | . . . . . . . 8 | |
34 | 1, 33 | syl 14 | . . . . . . 7 |
35 | 30, 32, 34 | rspcdva 2794 | . . . . . 6 |
36 | eluzel2 9331 | . . . . . . . 8 | |
37 | 1, 36 | syl 14 | . . . . . . 7 |
38 | seq3shft2.f | . . . . . . 7 | |
39 | seq3shft2.pl | . . . . . . 7 | |
40 | 37, 38, 39 | seq3-1 10233 | . . . . . 6 |
41 | seq3shft2.2 | . . . . . . . 8 | |
42 | 37, 41 | zaddcld 9177 | . . . . . . 7 |
43 | seq3shft2.g | . . . . . . 7 | |
44 | 42, 43, 39 | seq3-1 10233 | . . . . . 6 |
45 | 35, 40, 44 | 3eqtr4d 2182 | . . . . 5 |
46 | 45 | a1i13 24 | . . . 4 |
47 | peano2fzr 9817 | . . . . . . . 8 | |
48 | 47 | adantl 275 | . . . . . . 7 |
49 | 48 | expr 372 | . . . . . 6 |
50 | 49 | imim1d 75 | . . . . 5 |
51 | oveq1 5781 | . . . . . 6 | |
52 | simprl 520 | . . . . . . . 8 | |
53 | 38 | adantlr 468 | . . . . . . . 8 |
54 | 39 | adantlr 468 | . . . . . . . 8 |
55 | 52, 53, 54 | seq3p1 10235 | . . . . . . 7 |
56 | 41 | adantr 274 | . . . . . . . . . 10 |
57 | eluzadd 9354 | . . . . . . . . . 10 | |
58 | 52, 56, 57 | syl2anc 408 | . . . . . . . . 9 |
59 | 43 | adantlr 468 | . . . . . . . . 9 |
60 | 58, 59, 54 | seq3p1 10235 | . . . . . . . 8 |
61 | eluzelz 9335 | . . . . . . . . . . . 12 | |
62 | 52, 61 | syl 14 | . . . . . . . . . . 11 |
63 | 62 | zcnd 9174 | . . . . . . . . . 10 |
64 | 1cnd 7782 | . . . . . . . . . 10 | |
65 | 56 | zcnd 9174 | . . . . . . . . . 10 |
66 | 63, 64, 65 | add32d 7930 | . . . . . . . . 9 |
67 | 66 | fveq2d 5425 | . . . . . . . 8 |
68 | fveq2 5421 | . . . . . . . . . . . 12 | |
69 | fvoveq1 5797 | . . . . . . . . . . . 12 | |
70 | 68, 69 | eqeq12d 2154 | . . . . . . . . . . 11 |
71 | 32 | adantr 274 | . . . . . . . . . . 11 |
72 | simprr 521 | . . . . . . . . . . 11 | |
73 | 70, 71, 72 | rspcdva 2794 | . . . . . . . . . 10 |
74 | 66 | fveq2d 5425 | . . . . . . . . . 10 |
75 | 73, 74 | eqtrd 2172 | . . . . . . . . 9 |
76 | 75 | oveq2d 5790 | . . . . . . . 8 |
77 | 60, 67, 76 | 3eqtr4d 2182 | . . . . . . 7 |
78 | 55, 77 | eqeq12d 2154 | . . . . . 6 |
79 | 51, 78 | syl5ibr 155 | . . . . 5 |
80 | 50, 79 | animpimp2impd 548 | . . . 4 |
81 | 9, 15, 21, 27, 46, 80 | uzind4 9383 | . . 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 1331 wcel 1480 wral 2416 cfv 5123 (class class class)co 5774 c1 7621 caddc 7623 cz 9054 cuz 9326 cfz 9790 cseq 10218 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-seqfrec 10219 |
This theorem is referenced by: seq3f1olemqsumkj 10271 seq3shft 10610 |
Copyright terms: Public domain | W3C validator |