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Mirrors > Home > ILE Home > Th. List > shftfn | Unicode version |
Description: Functionality and domain of a sequence shifted by . (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
shftfval.1 |
Ref | Expression |
---|---|
shftfn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 4464 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | fnfun 4996 | . . . . . 6 | |
4 | 3 | adantr 261 | . . . . 5 |
5 | funmo 4917 | . . . . . . 7 | |
6 | vex 2560 | . . . . . . . . . 10 | |
7 | vex 2560 | . . . . . . . . . 10 | |
8 | eleq1 2100 | . . . . . . . . . . 11 | |
9 | oveq1 5519 | . . . . . . . . . . . 12 | |
10 | 9 | breq1d 3774 | . . . . . . . . . . 11 |
11 | 8, 10 | anbi12d 442 | . . . . . . . . . 10 |
12 | breq2 3768 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 437 | . . . . . . . . . 10 |
14 | eqid 2040 | . . . . . . . . . 10 | |
15 | 6, 7, 11, 13, 14 | brab 4009 | . . . . . . . . 9 |
16 | 15 | simprbi 260 | . . . . . . . 8 |
17 | 16 | moimi 1965 | . . . . . . 7 |
18 | 5, 17 | syl 14 | . . . . . 6 |
19 | 18 | alrimiv 1754 | . . . . 5 |
20 | 4, 19 | syl 14 | . . . 4 |
21 | dffun6 4916 | . . . 4 | |
22 | 2, 20, 21 | sylanbrc 394 | . . 3 |
23 | shftfval.1 | . . . . . 6 | |
24 | 23 | shftfval 9422 | . . . . 5 |
25 | 24 | adantl 262 | . . . 4 |
26 | 25 | funeqd 4923 | . . 3 |
27 | 22, 26 | mpbird 156 | . 2 |
28 | 23 | shftdm 9423 | . . 3 |
29 | fndm 4998 | . . . . 5 | |
30 | 29 | eleq2d 2107 | . . . 4 |
31 | 30 | rabbidv 2549 | . . 3 |
32 | 28, 31 | sylan9eqr 2094 | . 2 |
33 | df-fn 4905 | . 2 | |
34 | 27, 32, 33 | sylanbrc 394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wal 1241 wceq 1243 wcel 1393 wmo 1901 crab 2310 cvv 2557 class class class wbr 3764 copab 3817 cdm 4345 wrel 4350 wfun 4896 wfn 4897 (class class class)co 5512 cc 6887 cmin 7182 cshi 9415 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-shft 9416 |
This theorem is referenced by: shftf 9431 |
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