Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > difelfznle | Unicode version |
Description: The difference of two integers from a finite set of sequential nonnegative integers increased by the upper bound is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.) |
Ref | Expression |
---|---|
difelfznle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2nn0 9847 | . . . . . 6 | |
2 | nn0addcl 8970 | . . . . . . . 8 | |
3 | 2 | nn0zd 9129 | . . . . . . 7 |
4 | 3 | 3adant3 986 | . . . . . 6 |
5 | 1, 4 | sylbi 120 | . . . . 5 |
6 | elfzelz 9761 | . . . . 5 | |
7 | zsubcl 9053 | . . . . 5 | |
8 | 5, 6, 7 | syl2anr 288 | . . . 4 |
9 | 8 | 3adant3 986 | . . 3 |
10 | 6 | zred 9131 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | elfzel2 9759 | . . . . . . . 8 | |
13 | 12 | zred 9131 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | nn0readdcl 8994 | . . . . . . . . 9 | |
16 | 15 | 3adant3 986 | . . . . . . . 8 |
17 | 1, 16 | sylbi 120 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | elfzle2 9763 | . . . . . . 7 | |
20 | elfzle1 9762 | . . . . . . . 8 | |
21 | nn0re 8944 | . . . . . . . . . . . 12 | |
22 | nn0re 8944 | . . . . . . . . . . . 12 | |
23 | 21, 22 | anim12ci 337 | . . . . . . . . . . 11 |
24 | 23 | 3adant3 986 | . . . . . . . . . 10 |
25 | 1, 24 | sylbi 120 | . . . . . . . . 9 |
26 | addge02 8203 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 20, 27 | mpbid 146 | . . . . . . 7 |
29 | 19, 28 | anim12i 336 | . . . . . 6 |
30 | letr 7815 | . . . . . . 7 | |
31 | 30 | imp 123 | . . . . . 6 |
32 | 11, 14, 18, 29, 31 | syl31anc 1204 | . . . . 5 |
33 | 32 | 3adant3 986 | . . . 4 |
34 | zre 9016 | . . . . . . . 8 | |
35 | 21, 22 | anim12i 336 | . . . . . . . . . . 11 |
36 | 35 | 3adant3 986 | . . . . . . . . . 10 |
37 | 1, 36 | sylbi 120 | . . . . . . . . 9 |
38 | readdcl 7714 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | 34, 39 | anim12ci 337 | . . . . . . 7 |
41 | 6, 40 | sylan 281 | . . . . . 6 |
42 | 41 | 3adant3 986 | . . . . 5 |
43 | subge0 8205 | . . . . 5 | |
44 | 42, 43 | syl 14 | . . . 4 |
45 | 33, 44 | mpbird 166 | . . 3 |
46 | elnn0z 9025 | . . 3 | |
47 | 9, 45, 46 | sylanbrc 413 | . 2 |
48 | elfz3nn0 9850 | . . 3 | |
49 | 48 | 3ad2ant1 987 | . 2 |
50 | elfzelz 9761 | . . . . . 6 | |
51 | zltnle 9058 | . . . . . . . 8 | |
52 | 51 | ancoms 266 | . . . . . . 7 |
53 | zre 9016 | . . . . . . . 8 | |
54 | ltle 7819 | . . . . . . . 8 | |
55 | 53, 34, 54 | syl2anr 288 | . . . . . . 7 |
56 | 52, 55 | sylbird 169 | . . . . . 6 |
57 | 6, 50, 56 | syl2an 287 | . . . . 5 |
58 | 57 | 3impia 1163 | . . . 4 |
59 | 50 | zred 9131 | . . . . . . 7 |
60 | 59 | adantl 275 | . . . . . 6 |
61 | 60, 11, 14 | leadd1d 8268 | . . . . 5 |
62 | 61 | 3adant3 986 | . . . 4 |
63 | 58, 62 | mpbid 146 | . . 3 |
64 | 18, 11, 14 | lesubadd2d 8273 | . . . 4 |
65 | 64 | 3adant3 986 | . . 3 |
66 | 63, 65 | mpbird 166 | . 2 |
67 | elfz2nn0 9847 | . 2 | |
68 | 47, 49, 66, 67 | syl3anbrc 1150 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 947 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 caddc 7591 clt 7768 cle 7769 cmin 7901 cn0 8935 cz 9012 cfz 9745 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-fz 9746 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |