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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 2164 | . 2 | |
2 | elfzelz 9951 | . . . 4 | |
3 | 2 | adantl 275 | . . 3 |
4 | mptfzshft.1 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | zsubcld 9309 | . 2 |
7 | elfzelz 9951 | . . . 4 | |
8 | 7 | adantl 275 | . . 3 |
9 | 4 | adantr 274 | . . 3 |
10 | 8, 9 | zaddcld 9308 | . 2 |
11 | simprr 522 | . . . . . . . 8 | |
12 | 11 | oveq1d 5851 | . . . . . . 7 |
13 | 2 | ad2antrl 482 | . . . . . . . 8 |
14 | 4 | adantr 274 | . . . . . . . 8 |
15 | zcn 9187 | . . . . . . . . 9 | |
16 | zcn 9187 | . . . . . . . . 9 | |
17 | npcan 8098 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2an 287 | . . . . . . . 8 |
19 | 13, 14, 18 | syl2anc 409 | . . . . . . 7 |
20 | 12, 19 | eqtr2d 2198 | . . . . . 6 |
21 | simprl 521 | . . . . . 6 | |
22 | 20, 21 | eqeltrrd 2242 | . . . . 5 |
23 | mptfzshft.2 | . . . . . . 7 | |
24 | 23 | adantr 274 | . . . . . 6 |
25 | mptfzshft.3 | . . . . . . 7 | |
26 | 25 | adantr 274 | . . . . . 6 |
27 | 13, 14 | zsubcld 9309 | . . . . . . 7 |
28 | 11, 27 | eqeltrd 2241 | . . . . . 6 |
29 | fzaddel 9984 | . . . . . 6 | |
30 | 24, 26, 28, 14, 29 | syl22anc 1228 | . . . . 5 |
31 | 22, 30 | mpbird 166 | . . . 4 |
32 | 31, 20 | jca 304 | . . 3 |
33 | simprr 522 | . . . . 5 | |
34 | simprl 521 | . . . . . 6 | |
35 | 23 | adantr 274 | . . . . . . 7 |
36 | 25 | adantr 274 | . . . . . . 7 |
37 | 7 | ad2antrl 482 | . . . . . . 7 |
38 | 4 | adantr 274 | . . . . . . 7 |
39 | 35, 36, 37, 38, 29 | syl22anc 1228 | . . . . . 6 |
40 | 34, 39 | mpbid 146 | . . . . 5 |
41 | 33, 40 | eqeltrd 2241 | . . . 4 |
42 | 33 | oveq1d 5851 | . . . . 5 |
43 | zcn 9187 | . . . . . . 7 | |
44 | pncan 8095 | . . . . . . 7 | |
45 | 43, 16, 44 | syl2an 287 | . . . . . 6 |
46 | 37, 38, 45 | syl2anc 409 | . . . . 5 |
47 | 42, 46 | eqtr2d 2198 | . . . 4 |
48 | 41, 47 | jca 304 | . . 3 |
49 | 32, 48 | impbida 586 | . 2 |
50 | 1, 6, 10, 49 | f1od 6035 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cmpt 4037 wf1o 5181 (class class class)co 5836 cc 7742 caddc 7747 cmin 8060 cz 9182 cfz 9935 |
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-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 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-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 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-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 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-int 3819 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-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: fsumshft 11371 fprodshft 11545 |
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