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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 9885 | . . . . . 6 | |
2 | nn0addcl 9005 | . . . . . . . 8 | |
3 | 2 | nn0zd 9164 | . . . . . . 7 |
4 | 3 | 3adant3 1001 | . . . . . 6 |
5 | 1, 4 | sylbi 120 | . . . . 5 |
6 | elfzelz 9799 | . . . . 5 | |
7 | zsubcl 9088 | . . . . 5 | |
8 | 5, 6, 7 | syl2anr 288 | . . . 4 |
9 | 8 | 3adant3 1001 | . . 3 |
10 | 6 | zred 9166 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | elfzel2 9797 | . . . . . . . 8 | |
13 | 12 | zred 9166 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | nn0readdcl 9029 | . . . . . . . . 9 | |
16 | 15 | 3adant3 1001 | . . . . . . . 8 |
17 | 1, 16 | sylbi 120 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | elfzle2 9801 | . . . . . . 7 | |
20 | elfzle1 9800 | . . . . . . . 8 | |
21 | nn0re 8979 | . . . . . . . . . . . 12 | |
22 | nn0re 8979 | . . . . . . . . . . . 12 | |
23 | 21, 22 | anim12ci 337 | . . . . . . . . . . 11 |
24 | 23 | 3adant3 1001 | . . . . . . . . . 10 |
25 | 1, 24 | sylbi 120 | . . . . . . . . 9 |
26 | addge02 8228 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 20, 27 | mpbid 146 | . . . . . . 7 |
29 | 19, 28 | anim12i 336 | . . . . . 6 |
30 | letr 7840 | . . . . . . 7 | |
31 | 30 | imp 123 | . . . . . 6 |
32 | 11, 14, 18, 29, 31 | syl31anc 1219 | . . . . 5 |
33 | 32 | 3adant3 1001 | . . . 4 |
34 | zre 9051 | . . . . . . . 8 | |
35 | 21, 22 | anim12i 336 | . . . . . . . . . . 11 |
36 | 35 | 3adant3 1001 | . . . . . . . . . 10 |
37 | 1, 36 | sylbi 120 | . . . . . . . . 9 |
38 | readdcl 7739 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | 34, 39 | anim12ci 337 | . . . . . . 7 |
41 | 6, 40 | sylan 281 | . . . . . 6 |
42 | 41 | 3adant3 1001 | . . . . 5 |
43 | subge0 8230 | . . . . 5 | |
44 | 42, 43 | syl 14 | . . . 4 |
45 | 33, 44 | mpbird 166 | . . 3 |
46 | elnn0z 9060 | . . 3 | |
47 | 9, 45, 46 | sylanbrc 413 | . 2 |
48 | elfz3nn0 9888 | . . 3 | |
49 | 48 | 3ad2ant1 1002 | . 2 |
50 | elfzelz 9799 | . . . . . 6 | |
51 | zltnle 9093 | . . . . . . . 8 | |
52 | 51 | ancoms 266 | . . . . . . 7 |
53 | zre 9051 | . . . . . . . 8 | |
54 | ltle 7844 | . . . . . . . 8 | |
55 | 53, 34, 54 | syl2anr 288 | . . . . . . 7 |
56 | 52, 55 | sylbird 169 | . . . . . 6 |
57 | 6, 50, 56 | syl2an 287 | . . . . 5 |
58 | 57 | 3impia 1178 | . . . 4 |
59 | 50 | zred 9166 | . . . . . . 7 |
60 | 59 | adantl 275 | . . . . . 6 |
61 | 60, 11, 14 | leadd1d 8294 | . . . . 5 |
62 | 61 | 3adant3 1001 | . . . 4 |
63 | 58, 62 | mpbid 146 | . . 3 |
64 | 18, 11, 14 | lesubadd2d 8299 | . . . 4 |
65 | 64 | 3adant3 1001 | . . 3 |
66 | 63, 65 | mpbird 166 | . 2 |
67 | elfz2nn0 9885 | . 2 | |
68 | 47, 49, 66, 67 | syl3anbrc 1165 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 cc0 7613 caddc 7616 clt 7793 cle 7794 cmin 7926 cn0 8970 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-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: (None) |
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