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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 9128 | . . . . . 6 | |
2 | nn0addcl 8323 | . . . . . . . 8 | |
3 | 2 | nn0zd 8467 | . . . . . . 7 |
4 | 3 | 3adant3 958 | . . . . . 6 |
5 | 1, 4 | sylbi 119 | . . . . 5 |
6 | elfzelz 9045 | . . . . 5 | |
7 | zsubcl 8392 | . . . . 5 | |
8 | 5, 6, 7 | syl2anr 284 | . . . 4 |
9 | 8 | 3adant3 958 | . . 3 |
10 | 6 | zred 8469 | . . . . . . 7 |
11 | 10 | adantr 270 | . . . . . 6 |
12 | elfzel2 9043 | . . . . . . . 8 | |
13 | 12 | zred 8469 | . . . . . . 7 |
14 | 13 | adantr 270 | . . . . . 6 |
15 | nn0readdcl 8347 | . . . . . . . . 9 | |
16 | 15 | 3adant3 958 | . . . . . . . 8 |
17 | 1, 16 | sylbi 119 | . . . . . . 7 |
18 | 17 | adantl 271 | . . . . . 6 |
19 | elfzle2 9047 | . . . . . . 7 | |
20 | elfzle1 9046 | . . . . . . . 8 | |
21 | nn0re 8297 | . . . . . . . . . . . 12 | |
22 | nn0re 8297 | . . . . . . . . . . . 12 | |
23 | 21, 22 | anim12ci 332 | . . . . . . . . . . 11 |
24 | 23 | 3adant3 958 | . . . . . . . . . 10 |
25 | 1, 24 | sylbi 119 | . . . . . . . . 9 |
26 | addge02 7577 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 20, 27 | mpbid 145 | . . . . . . 7 |
29 | 19, 28 | anim12i 331 | . . . . . 6 |
30 | letr 7194 | . . . . . . 7 | |
31 | 30 | imp 122 | . . . . . 6 |
32 | 11, 14, 18, 29, 31 | syl31anc 1172 | . . . . 5 |
33 | 32 | 3adant3 958 | . . . 4 |
34 | zre 8355 | . . . . . . . 8 | |
35 | 21, 22 | anim12i 331 | . . . . . . . . . . 11 |
36 | 35 | 3adant3 958 | . . . . . . . . . 10 |
37 | 1, 36 | sylbi 119 | . . . . . . . . 9 |
38 | readdcl 7099 | . . . . . . . . 9 | |
39 | 37, 38 | syl 14 | . . . . . . . 8 |
40 | 34, 39 | anim12ci 332 | . . . . . . 7 |
41 | 6, 40 | sylan 277 | . . . . . 6 |
42 | 41 | 3adant3 958 | . . . . 5 |
43 | subge0 7579 | . . . . 5 | |
44 | 42, 43 | syl 14 | . . . 4 |
45 | 33, 44 | mpbird 165 | . . 3 |
46 | elnn0z 8364 | . . 3 | |
47 | 9, 45, 46 | sylanbrc 408 | . 2 |
48 | elfz3nn0 9131 | . . 3 | |
49 | 48 | 3ad2ant1 959 | . 2 |
50 | elfzelz 9045 | . . . . . 6 | |
51 | zltnle 8397 | . . . . . . . 8 | |
52 | 51 | ancoms 264 | . . . . . . 7 |
53 | zre 8355 | . . . . . . . 8 | |
54 | ltle 7198 | . . . . . . . 8 | |
55 | 53, 34, 54 | syl2anr 284 | . . . . . . 7 |
56 | 52, 55 | sylbird 168 | . . . . . 6 |
57 | 6, 50, 56 | syl2an 283 | . . . . 5 |
58 | 57 | 3impia 1135 | . . . 4 |
59 | 50 | zred 8469 | . . . . . . 7 |
60 | 59 | adantl 271 | . . . . . 6 |
61 | 60, 11, 14 | leadd1d 7639 | . . . . 5 |
62 | 61 | 3adant3 958 | . . . 4 |
63 | 58, 62 | mpbid 145 | . . 3 |
64 | 18, 11, 14 | lesubadd2d 7644 | . . . 4 |
65 | 64 | 3adant3 958 | . . 3 |
66 | 63, 65 | mpbird 165 | . 2 |
67 | elfz2nn0 9128 | . 2 | |
68 | 47, 49, 66, 67 | syl3anbrc 1122 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 caddc 6984 clt 7153 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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