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Mirrors > Home > ILE Home > Th. List > fz0fzdiffz0 | Unicode version |
Description: The difference of an integer in a finite set of sequential nonnegative integers and and an integer of a finite set of sequential integers with the same upper bound and the nonnegative integer as lower bound is a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 6-Jun-2018.) |
Ref | Expression |
---|---|
fz0fzdiffz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz0fzelfz0 9904 | . . 3 | |
2 | elfzle1 9807 | . . . . . . 7 | |
3 | 2 | adantl 275 | . . . . . 6 |
4 | 3 | adantl 275 | . . . . 5 |
5 | elfznn0 9894 | . . . . . . 7 | |
6 | 5 | adantr 274 | . . . . . 6 |
7 | elfznn0 9894 | . . . . . 6 | |
8 | nn0sub 9120 | . . . . . 6 | |
9 | 6, 7, 8 | syl2anr 288 | . . . . 5 |
10 | 4, 9 | mpbid 146 | . . . 4 |
11 | elfz3nn0 9895 | . . . . 5 | |
12 | 11 | adantr 274 | . . . 4 |
13 | elfz2nn0 9892 | . . . . . . 7 | |
14 | elfz2 9797 | . . . . . . . . . . 11 | |
15 | zsubcl 9095 | . . . . . . . . . . . . . . . . . . . . . . 23 | |
16 | 15 | zred 9173 | . . . . . . . . . . . . . . . . . . . . . 22 |
17 | 16 | ancoms 266 | . . . . . . . . . . . . . . . . . . . . 21 |
18 | 17 | 3adant2 1000 | . . . . . . . . . . . . . . . . . . . 20 |
19 | zre 9058 | . . . . . . . . . . . . . . . . . . . . 21 | |
20 | 19 | 3ad2ant3 1004 | . . . . . . . . . . . . . . . . . . . 20 |
21 | zre 9058 | . . . . . . . . . . . . . . . . . . . . 21 | |
22 | 21 | 3ad2ant2 1003 | . . . . . . . . . . . . . . . . . . . 20 |
23 | 18, 20, 22 | 3jca 1161 | . . . . . . . . . . . . . . . . . . 19 |
24 | 23 | adantr 274 | . . . . . . . . . . . . . . . . . 18 |
25 | 24 | adantr 274 | . . . . . . . . . . . . . . . . 17 |
26 | nn0ge0 9002 | . . . . . . . . . . . . . . . . . . . 20 | |
27 | 26 | adantl 275 | . . . . . . . . . . . . . . . . . . 19 |
28 | nn0re 8986 | . . . . . . . . . . . . . . . . . . . 20 | |
29 | subge02 8240 | . . . . . . . . . . . . . . . . . . . 20 | |
30 | 20, 28, 29 | syl2an 287 | . . . . . . . . . . . . . . . . . . 19 |
31 | 27, 30 | mpbid 146 | . . . . . . . . . . . . . . . . . 18 |
32 | 31 | anim1i 338 | . . . . . . . . . . . . . . . . 17 |
33 | letr 7847 | . . . . . . . . . . . . . . . . 17 | |
34 | 25, 32, 33 | sylc 62 | . . . . . . . . . . . . . . . 16 |
35 | 34 | exp31 361 | . . . . . . . . . . . . . . 15 |
36 | 35 | a1i 9 | . . . . . . . . . . . . . 14 |
37 | 36 | com14 88 | . . . . . . . . . . . . 13 |
38 | 37 | adantl 275 | . . . . . . . . . . . 12 |
39 | 38 | impcom 124 | . . . . . . . . . . 11 |
40 | 14, 39 | sylbi 120 | . . . . . . . . . 10 |
41 | 40 | com13 80 | . . . . . . . . 9 |
42 | 41 | impcom 124 | . . . . . . . 8 |
43 | 42 | 3adant3 1001 | . . . . . . 7 |
44 | 13, 43 | sylbi 120 | . . . . . 6 |
45 | 44 | imp 123 | . . . . 5 |
46 | 45 | adantl 275 | . . . 4 |
47 | 10, 12, 46 | 3jca 1161 | . . 3 |
48 | 1, 47 | mpancom 418 | . 2 |
49 | elfz2nn0 9892 | . 2 | |
50 | 48, 49 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 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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