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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 9892 | . . . . . 6 | |
2 | nn0addcl 9012 | . . . . . . . 8 | |
3 | 2 | nn0zd 9171 | . . . . . . 7 |
4 | 3 | 3adant3 1001 | . . . . . 6 |
5 | 1, 4 | sylbi 120 | . . . . 5 |
6 | elfzelz 9806 | . . . . 5 | |
7 | zsubcl 9095 | . . . . 5 | |
8 | 5, 6, 7 | syl2anr 288 | . . . 4 |
9 | 8 | 3adant3 1001 | . . 3 |
10 | 6 | zred 9173 | . . . . . . 7 |
11 | 10 | adantr 274 | . . . . . 6 |
12 | elfzel2 9804 | . . . . . . . 8 | |
13 | 12 | zred 9173 | . . . . . . 7 |
14 | 13 | adantr 274 | . . . . . 6 |
15 | nn0readdcl 9036 | . . . . . . . . 9 | |
16 | 15 | 3adant3 1001 | . . . . . . . 8 |
17 | 1, 16 | sylbi 120 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | elfzle2 9808 | . . . . . . 7 | |
20 | elfzle1 9807 | . . . . . . . 8 | |
21 | nn0re 8986 | . . . . . . . . . . . 12 | |
22 | nn0re 8986 | . . . . . . . . . . . 12 | |
23 | 21, 22 | anim12ci 337 | . . . . . . . . . . 11 |
24 | 23 | 3adant3 1001 | . . . . . . . . . 10 |
25 | 1, 24 | sylbi 120 | . . . . . . . . 9 |
26 | addge02 8235 | . . . . . . . . 9 | |
27 | 25, 26 | syl 14 | . . . . . . . 8 |
28 | 20, 27 | mpbid 146 | . . . . . . 7 |
29 | 19, 28 | anim12i 336 | . . . . . 6 |
30 | letr 7847 | . . . . . . 7 | |
31 | 30 | imp 123 | . . . . . 6 |
32 | 11, 14, 18, 29, 31 | syl31anc 1219 | . . . . 5 |
33 | 32 | 3adant3 1001 | . . . 4 |
34 | zre 9058 | . . . . . . . 8 | |
35 | 21, 22 | anim12i 336 | . . . . . . . . . . 11 |
36 | 35 | 3adant3 1001 | . . . . . . . . . 10 |
37 | 1, 36 | sylbi 120 | . . . . . . . . 9 |
38 | readdcl 7746 | . . . . . . . . 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 8237 | . . . . 5 | |
44 | 42, 43 | syl 14 | . . . 4 |
45 | 33, 44 | mpbird 166 | . . 3 |
46 | elnn0z 9067 | . . 3 | |
47 | 9, 45, 46 | sylanbrc 413 | . 2 |
48 | elfz3nn0 9895 | . . 3 | |
49 | 48 | 3ad2ant1 1002 | . 2 |
50 | elfzelz 9806 | . . . . . 6 | |
51 | zltnle 9100 | . . . . . . . 8 | |
52 | 51 | ancoms 266 | . . . . . . 7 |
53 | zre 9058 | . . . . . . . 8 | |
54 | ltle 7851 | . . . . . . . 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 9173 | . . . . . . 7 |
60 | 59 | adantl 275 | . . . . . 6 |
61 | 60, 11, 14 | leadd1d 8301 | . . . . 5 |
62 | 61 | 3adant3 1001 | . . . 4 |
63 | 58, 62 | mpbid 146 | . . 3 |
64 | 18, 11, 14 | lesubadd2d 8306 | . . . 4 |
65 | 64 | 3adant3 1001 | . . 3 |
66 | 63, 65 | mpbird 166 | . 2 |
67 | elfz2nn0 9892 | . 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 3929 (class class class)co 5774 cr 7619 cc0 7620 caddc 7623 clt 7800 cle 7801 cmin 7933 cn0 8977 cz 9054 cfz 9790 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
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