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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 2137 | . 2 | |
2 | elfzelz 9799 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | mptfzshft.1 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | zsubcld 9171 | . 2 |
7 | elfzelz 9799 | . . . 4 | |
8 | 7 | adantl 275 | . . 3 |
9 | 4 | adantr 274 | . . 3 |
10 | 8, 9 | zaddcld 9170 | . 2 |
11 | simprr 521 | . . . . . . . 8 | |
12 | 11 | oveq1d 5782 | . . . . . . 7 |
13 | 2 | ad2antrl 481 | . . . . . . . 8 |
14 | 4 | adantr 274 | . . . . . . . 8 |
15 | zcn 9052 | . . . . . . . . 9 | |
16 | zcn 9052 | . . . . . . . . 9 | |
17 | npcan 7964 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2an 287 | . . . . . . . 8 |
19 | 13, 14, 18 | syl2anc 408 | . . . . . . 7 |
20 | 12, 19 | eqtr2d 2171 | . . . . . 6 |
21 | simprl 520 | . . . . . 6 | |
22 | 20, 21 | eqeltrrd 2215 | . . . . 5 |
23 | mptfzshft.2 | . . . . . . 7 | |
24 | 23 | adantr 274 | . . . . . 6 |
25 | mptfzshft.3 | . . . . . . 7 | |
26 | 25 | adantr 274 | . . . . . 6 |
27 | 13, 14 | zsubcld 9171 | . . . . . . 7 |
28 | 11, 27 | eqeltrd 2214 | . . . . . 6 |
29 | fzaddel 9832 | . . . . . 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 2214 | . . . 4 |
42 | 33 | oveq1d 5782 | . . . . 5 |
43 | zcn 9052 | . . . . . . 7 | |
44 | pncan 7961 | . . . . . . 7 | |
45 | 43, 16, 44 | syl2an 287 | . . . . . 6 |
46 | 37, 38, 45 | syl2anc 408 | . . . . 5 |
47 | 42, 46 | eqtr2d 2171 | . . . 4 |
48 | 41, 47 | jca 304 | . . 3 |
49 | 32, 48 | impbida 585 | . 2 |
50 | 1, 6, 10, 49 | f1od 5966 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 cmpt 3984 wf1o 5117 (class class class)co 5767 cc 7611 caddc 7616 cmin 7926 cz 9047 cfz 9783 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 |
This theorem is referenced by: fsumshft 11206 |
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