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Mirrors > Home > ILE Home > Th. List > fz0fzdiffz0 | Unicode version |
Description: The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.) |
Ref | Expression |
---|---|
fz0fzdiffz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz0fzelfz0 8984 | . . 3 | |
2 | elfzle1 8891 | . . . . . . 7 | |
3 | 2 | adantl 262 | . . . . . 6 |
4 | 3 | adantl 262 | . . . . 5 |
5 | elfznn0 8975 | . . . . . . 7 | |
6 | 5 | adantr 261 | . . . . . 6 |
7 | elfznn0 8975 | . . . . . 6 | |
8 | nn0sub 8310 | . . . . . 6 | |
9 | 6, 7, 8 | syl2anr 274 | . . . . 5 |
10 | 4, 9 | mpbid 135 | . . . 4 |
11 | elfz3nn0 8976 | . . . . 5 | |
12 | 11 | adantr 261 | . . . 4 |
13 | elfz2nn0 8973 | . . . . . . 7 | |
14 | elfz2 8881 | . . . . . . . . . . 11 | |
15 | zsubcl 8286 | . . . . . . . . . . . . . . . . . . . . . . 23 | |
16 | 15 | zred 8360 | . . . . . . . . . . . . . . . . . . . . . 22 |
17 | 16 | ancoms 255 | . . . . . . . . . . . . . . . . . . . . 21 |
18 | 17 | 3adant2 923 | . . . . . . . . . . . . . . . . . . . 20 |
19 | zre 8249 | . . . . . . . . . . . . . . . . . . . . 21 | |
20 | 19 | 3ad2ant3 927 | . . . . . . . . . . . . . . . . . . . 20 |
21 | zre 8249 | . . . . . . . . . . . . . . . . . . . . 21 | |
22 | 21 | 3ad2ant2 926 | . . . . . . . . . . . . . . . . . . . 20 |
23 | 18, 20, 22 | 3jca 1084 | . . . . . . . . . . . . . . . . . . 19 |
24 | 23 | adantr 261 | . . . . . . . . . . . . . . . . . 18 |
25 | 24 | adantr 261 | . . . . . . . . . . . . . . . . 17 |
26 | nn0ge0 8207 | . . . . . . . . . . . . . . . . . . . 20 | |
27 | 26 | adantl 262 | . . . . . . . . . . . . . . . . . . 19 |
28 | nn0re 8190 | . . . . . . . . . . . . . . . . . . . 20 | |
29 | subge02 7473 | . . . . . . . . . . . . . . . . . . . 20 | |
30 | 20, 28, 29 | syl2an 273 | . . . . . . . . . . . . . . . . . . 19 |
31 | 27, 30 | mpbid 135 | . . . . . . . . . . . . . . . . . 18 |
32 | 31 | anim1i 323 | . . . . . . . . . . . . . . . . 17 |
33 | letr 7101 | . . . . . . . . . . . . . . . . 17 | |
34 | 25, 32, 33 | sylc 56 | . . . . . . . . . . . . . . . 16 |
35 | 34 | exp31 346 | . . . . . . . . . . . . . . 15 |
36 | 35 | a1i 9 | . . . . . . . . . . . . . 14 |
37 | 36 | com14 82 | . . . . . . . . . . . . 13 |
38 | 37 | adantl 262 | . . . . . . . . . . . 12 |
39 | 38 | impcom 116 | . . . . . . . . . . 11 |
40 | 14, 39 | sylbi 114 | . . . . . . . . . 10 |
41 | 40 | com13 74 | . . . . . . . . 9 |
42 | 41 | impcom 116 | . . . . . . . 8 |
43 | 42 | 3adant3 924 | . . . . . . 7 |
44 | 13, 43 | sylbi 114 | . . . . . 6 |
45 | 44 | imp 115 | . . . . 5 |
46 | 45 | adantl 262 | . . . 4 |
47 | 10, 12, 46 | 3jca 1084 | . . 3 |
48 | 1, 47 | mpancom 399 | . 2 |
49 | elfz2nn0 8973 | . 2 | |
50 | 48, 49 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cc0 6889 cle 7061 cmin 7182 cn0 8181 cz 8245 cfz 8874 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-fz 8875 |
This theorem is referenced by: (None) |
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