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Mirrors > Home > ILE Home > Th. List > difelfzle | Unicode version |
Description: The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.) |
Ref | Expression |
---|---|
difelfzle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 9130 | . . . . 5 | |
2 | elfznn0 9130 | . . . . 5 | |
3 | nn0z 8371 | . . . . . . . . 9 | |
4 | nn0z 8371 | . . . . . . . . 9 | |
5 | zsubcl 8392 | . . . . . . . . 9 | |
6 | 3, 4, 5 | syl2anr 284 | . . . . . . . 8 |
7 | 6 | adantr 270 | . . . . . . 7 |
8 | nn0re 8297 | . . . . . . . . 9 | |
9 | nn0re 8297 | . . . . . . . . 9 | |
10 | subge0 7579 | . . . . . . . . 9 | |
11 | 8, 9, 10 | syl2anr 284 | . . . . . . . 8 |
12 | 11 | biimpar 291 | . . . . . . 7 |
13 | 7, 12 | jca 300 | . . . . . 6 |
14 | 13 | exp31 356 | . . . . 5 |
15 | 1, 2, 14 | syl2im 38 | . . . 4 |
16 | 15 | 3imp 1132 | . . 3 |
17 | elnn0z 8364 | . . 3 | |
18 | 16, 17 | sylibr 132 | . 2 |
19 | elfz3nn0 9131 | . . 3 | |
20 | 19 | 3ad2ant1 959 | . 2 |
21 | elfz2nn0 9128 | . . . . . 6 | |
22 | 8 | 3ad2ant1 959 | . . . . . . . . 9 |
23 | resubcl 7372 | . . . . . . . . 9 | |
24 | 22, 9, 23 | syl2an 283 | . . . . . . . 8 |
25 | 22 | adantr 270 | . . . . . . . 8 |
26 | nn0re 8297 | . . . . . . . . . 10 | |
27 | 26 | 3ad2ant2 960 | . . . . . . . . 9 |
28 | 27 | adantr 270 | . . . . . . . 8 |
29 | nn0ge0 8313 | . . . . . . . . . 10 | |
30 | 29 | adantl 271 | . . . . . . . . 9 |
31 | subge02 7582 | . . . . . . . . . 10 | |
32 | 22, 9, 31 | syl2an 283 | . . . . . . . . 9 |
33 | 30, 32 | mpbid 145 | . . . . . . . 8 |
34 | simpl3 943 | . . . . . . . 8 | |
35 | 24, 25, 28, 33, 34 | letrd 7233 | . . . . . . 7 |
36 | 35 | ex 113 | . . . . . 6 |
37 | 21, 36 | sylbi 119 | . . . . 5 |
38 | 1, 37 | syl5com 29 | . . . 4 |
39 | 38 | a1dd 47 | . . 3 |
40 | 39 | 3imp 1132 | . 2 |
41 | elfz2nn0 9128 | . 2 | |
42 | 18, 20, 40, 41 | syl3anbrc 1122 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 cle 7154 cmin 7279 cn0 8288 cz 8351 cfz 9029 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 |
This theorem is referenced by: (None) |
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