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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 4636 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | fnfun 5190 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | funmo 5108 | . . . . . . 7 | |
6 | vex 2663 | . . . . . . . . . 10 | |
7 | vex 2663 | . . . . . . . . . 10 | |
8 | eleq1 2180 | . . . . . . . . . . 11 | |
9 | oveq1 5749 | . . . . . . . . . . . 12 | |
10 | 9 | breq1d 3909 | . . . . . . . . . . 11 |
11 | 8, 10 | anbi12d 464 | . . . . . . . . . 10 |
12 | breq2 3903 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 459 | . . . . . . . . . 10 |
14 | eqid 2117 | . . . . . . . . . 10 | |
15 | 6, 7, 11, 13, 14 | brab 4164 | . . . . . . . . 9 |
16 | 15 | simprbi 273 | . . . . . . . 8 |
17 | 16 | moimi 2042 | . . . . . . 7 |
18 | 5, 17 | syl 14 | . . . . . 6 |
19 | 18 | alrimiv 1830 | . . . . 5 |
20 | 4, 19 | syl 14 | . . . 4 |
21 | dffun6 5107 | . . . 4 | |
22 | 2, 20, 21 | sylanbrc 413 | . . 3 |
23 | shftfval.1 | . . . . . 6 | |
24 | 23 | shftfval 10548 | . . . . 5 |
25 | 24 | adantl 275 | . . . 4 |
26 | 25 | funeqd 5115 | . . 3 |
27 | 22, 26 | mpbird 166 | . 2 |
28 | 23 | shftdm 10549 | . . 3 |
29 | fndm 5192 | . . . . 5 | |
30 | 29 | eleq2d 2187 | . . . 4 |
31 | 30 | rabbidv 2649 | . . 3 |
32 | 28, 31 | sylan9eqr 2172 | . 2 |
33 | df-fn 5096 | . 2 | |
34 | 27, 32, 33 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1314 wceq 1316 wcel 1465 wmo 1978 crab 2397 cvv 2660 class class class wbr 3899 copab 3958 cdm 4509 wrel 4514 wfun 5087 wfn 5088 (class class class)co 5742 cc 7586 cmin 7901 cshi 10541 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-shft 10542 |
This theorem is referenced by: shftf 10557 seq3shft 10565 |
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