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Mirrors > Home > ILE Home > Th. List > mptfzshft | Unicode version |
Description: 1-1 onto function in maps-to notation which shifts a finite set of sequential integers. (Contributed by AV, 24-Aug-2019.) |
Ref | Expression |
---|---|
mptfzshft.1 | |
mptfzshft.2 | |
mptfzshft.3 |
Ref | Expression |
---|---|
mptfzshft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . 2 | |
2 | elfzelz 9809 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | mptfzshft.1 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | zsubcld 9181 | . 2 |
7 | elfzelz 9809 | . . . 4 | |
8 | 7 | adantl 275 | . . 3 |
9 | 4 | adantr 274 | . . 3 |
10 | 8, 9 | zaddcld 9180 | . 2 |
11 | simprr 521 | . . . . . . . 8 | |
12 | 11 | oveq1d 5789 | . . . . . . 7 |
13 | 2 | ad2antrl 481 | . . . . . . . 8 |
14 | 4 | adantr 274 | . . . . . . . 8 |
15 | zcn 9062 | . . . . . . . . 9 | |
16 | zcn 9062 | . . . . . . . . 9 | |
17 | npcan 7974 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2an 287 | . . . . . . . 8 |
19 | 13, 14, 18 | syl2anc 408 | . . . . . . 7 |
20 | 12, 19 | eqtr2d 2173 | . . . . . 6 |
21 | simprl 520 | . . . . . 6 | |
22 | 20, 21 | eqeltrrd 2217 | . . . . 5 |
23 | mptfzshft.2 | . . . . . . 7 | |
24 | 23 | adantr 274 | . . . . . 6 |
25 | mptfzshft.3 | . . . . . . 7 | |
26 | 25 | adantr 274 | . . . . . 6 |
27 | 13, 14 | zsubcld 9181 | . . . . . . 7 |
28 | 11, 27 | eqeltrd 2216 | . . . . . 6 |
29 | fzaddel 9842 | . . . . . 6 | |
30 | 24, 26, 28, 14, 29 | syl22anc 1217 | . . . . 5 |
31 | 22, 30 | mpbird 166 | . . . 4 |
32 | 31, 20 | jca 304 | . . 3 |
33 | simprr 521 | . . . . 5 | |
34 | simprl 520 | . . . . . 6 | |
35 | 23 | adantr 274 | . . . . . . 7 |
36 | 25 | adantr 274 | . . . . . . 7 |
37 | 7 | ad2antrl 481 | . . . . . . 7 |
38 | 4 | adantr 274 | . . . . . . 7 |
39 | 35, 36, 37, 38, 29 | syl22anc 1217 | . . . . . 6 |
40 | 34, 39 | mpbid 146 | . . . . 5 |
41 | 33, 40 | eqeltrd 2216 | . . . 4 |
42 | 33 | oveq1d 5789 | . . . . 5 |
43 | zcn 9062 | . . . . . . 7 | |
44 | pncan 7971 | . . . . . . 7 | |
45 | 43, 16, 44 | syl2an 287 | . . . . . 6 |
46 | 37, 38, 45 | syl2anc 408 | . . . . 5 |
47 | 42, 46 | eqtr2d 2173 | . . . 4 |
48 | 41, 47 | jca 304 | . . 3 |
49 | 32, 48 | impbida 585 | . 2 |
50 | 1, 6, 10, 49 | f1od 5973 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cmpt 3989 wf1o 5122 (class class class)co 5774 cc 7621 caddc 7626 cmin 7936 cz 9057 cfz 9793 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 |
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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 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-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 df-uz 9330 df-fz 9794 |
This theorem is referenced by: fsumshft 11216 |
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