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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 10076 | . . 3 | |
2 | elfzle1 9976 | . . . . . . 7 | |
3 | 2 | adantl 275 | . . . . . 6 |
4 | 3 | adantl 275 | . . . . 5 |
5 | elfznn0 10063 | . . . . . . 7 | |
6 | 5 | adantr 274 | . . . . . 6 |
7 | elfznn0 10063 | . . . . . 6 | |
8 | nn0sub 9271 | . . . . . 6 | |
9 | 6, 7, 8 | syl2anr 288 | . . . . 5 |
10 | 4, 9 | mpbid 146 | . . . 4 |
11 | elfz3nn0 10064 | . . . . 5 | |
12 | 11 | adantr 274 | . . . 4 |
13 | elfz2nn0 10061 | . . . . . . 7 | |
14 | elfz2 9965 | . . . . . . . . . . 11 | |
15 | zsubcl 9246 | . . . . . . . . . . . . . . . . . . . . . . 23 | |
16 | 15 | zred 9327 | . . . . . . . . . . . . . . . . . . . . . 22 |
17 | 16 | ancoms 266 | . . . . . . . . . . . . . . . . . . . . 21 |
18 | 17 | 3adant2 1011 | . . . . . . . . . . . . . . . . . . . 20 |
19 | zre 9209 | . . . . . . . . . . . . . . . . . . . . 21 | |
20 | 19 | 3ad2ant3 1015 | . . . . . . . . . . . . . . . . . . . 20 |
21 | zre 9209 | . . . . . . . . . . . . . . . . . . . . 21 | |
22 | 21 | 3ad2ant2 1014 | . . . . . . . . . . . . . . . . . . . 20 |
23 | 18, 20, 22 | 3jca 1172 | . . . . . . . . . . . . . . . . . . 19 |
24 | 23 | adantr 274 | . . . . . . . . . . . . . . . . . 18 |
25 | 24 | adantr 274 | . . . . . . . . . . . . . . . . 17 |
26 | nn0ge0 9153 | . . . . . . . . . . . . . . . . . . . 20 | |
27 | 26 | adantl 275 | . . . . . . . . . . . . . . . . . . 19 |
28 | nn0re 9137 | . . . . . . . . . . . . . . . . . . . 20 | |
29 | subge02 8390 | . . . . . . . . . . . . . . . . . . . 20 | |
30 | 20, 28, 29 | syl2an 287 | . . . . . . . . . . . . . . . . . . 19 |
31 | 27, 30 | mpbid 146 | . . . . . . . . . . . . . . . . . 18 |
32 | 31 | anim1i 338 | . . . . . . . . . . . . . . . . 17 |
33 | letr 7995 | . . . . . . . . . . . . . . . . 17 | |
34 | 25, 32, 33 | sylc 62 | . . . . . . . . . . . . . . . 16 |
35 | 34 | exp31 362 | . . . . . . . . . . . . . . 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 1012 | . . . . . . 7 |
44 | 13, 43 | sylbi 120 | . . . . . 6 |
45 | 44 | imp 123 | . . . . 5 |
46 | 45 | adantl 275 | . . . 4 |
47 | 10, 12, 46 | 3jca 1172 | . . 3 |
48 | 1, 47 | mpancom 420 | . 2 |
49 | elfz2nn0 10061 | . 2 | |
50 | 48, 49 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wcel 2141 class class class wbr 3987 (class class class)co 5851 cr 7766 cc0 7767 cle 7948 cmin 8083 cn0 9128 cz 9205 cfz 9958 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-ltadd 7883 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-inn 8872 df-n0 9129 df-z 9206 df-uz 9481 df-fz 9959 |
This theorem is referenced by: (None) |
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