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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 4725 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | fnfun 5279 | . . . . . 6 | |
4 | 3 | adantr 274 | . . . . 5 |
5 | funmo 5197 | . . . . . . 7 | |
6 | vex 2724 | . . . . . . . . . 10 | |
7 | vex 2724 | . . . . . . . . . 10 | |
8 | eleq1 2227 | . . . . . . . . . . 11 | |
9 | oveq1 5843 | . . . . . . . . . . . 12 | |
10 | 9 | breq1d 3986 | . . . . . . . . . . 11 |
11 | 8, 10 | anbi12d 465 | . . . . . . . . . 10 |
12 | breq2 3980 | . . . . . . . . . . 11 | |
13 | 12 | anbi2d 460 | . . . . . . . . . 10 |
14 | eqid 2164 | . . . . . . . . . 10 | |
15 | 6, 7, 11, 13, 14 | brab 4244 | . . . . . . . . 9 |
16 | 15 | simprbi 273 | . . . . . . . 8 |
17 | 16 | moimi 2078 | . . . . . . 7 |
18 | 5, 17 | syl 14 | . . . . . 6 |
19 | 18 | alrimiv 1861 | . . . . 5 |
20 | 4, 19 | syl 14 | . . . 4 |
21 | dffun6 5196 | . . . 4 | |
22 | 2, 20, 21 | sylanbrc 414 | . . 3 |
23 | shftfval.1 | . . . . . 6 | |
24 | 23 | shftfval 10749 | . . . . 5 |
25 | 24 | adantl 275 | . . . 4 |
26 | 25 | funeqd 5204 | . . 3 |
27 | 22, 26 | mpbird 166 | . 2 |
28 | 23 | shftdm 10750 | . . 3 |
29 | fndm 5281 | . . . . 5 | |
30 | 29 | eleq2d 2234 | . . . 4 |
31 | 30 | rabbidv 2710 | . . 3 |
32 | 28, 31 | sylan9eqr 2219 | . 2 |
33 | df-fn 5185 | . 2 | |
34 | 27, 32, 33 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1340 wceq 1342 wmo 2014 wcel 2135 crab 2446 cvv 2721 class class class wbr 3976 copab 4036 cdm 4598 wrel 4603 wfun 5176 wfn 5177 (class class class)co 5836 cc 7742 cmin 8060 cshi 10742 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-shft 10743 |
This theorem is referenced by: shftf 10758 seq3shft 10766 |
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