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Mirrors > Home > ILE Home > Th. List > zdcle | Unicode version |
Description: Integer is decidable. (Contributed by Jim Kingdon, 7-Apr-2020.) |
Ref | Expression |
---|---|
zdcle | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ztri3or 9100 | . 2 | |
2 | zre 9061 | . . 3 | |
3 | zre 9061 | . . 3 | |
4 | ltle 7854 | . . . . 5 | |
5 | orc 701 | . . . . . 6 | |
6 | df-dc 820 | . . . . . 6 DECID | |
7 | 5, 6 | sylibr 133 | . . . . 5 DECID |
8 | 4, 7 | syl6 33 | . . . 4 DECID |
9 | eqle 7858 | . . . . . . 7 | |
10 | 9, 7 | syl 14 | . . . . . 6 DECID |
11 | 10 | ex 114 | . . . . 5 DECID |
12 | 11 | adantr 274 | . . . 4 DECID |
13 | lenlt 7843 | . . . . . . 7 | |
14 | 13 | biimpd 143 | . . . . . 6 |
15 | 14 | con2d 613 | . . . . 5 |
16 | olc 700 | . . . . . 6 | |
17 | 16, 6 | sylibr 133 | . . . . 5 DECID |
18 | 15, 17 | syl6 33 | . . . 4 DECID |
19 | 8, 12, 18 | 3jaod 1282 | . . 3 DECID |
20 | 2, 3, 19 | syl2an 287 | . 2 DECID |
21 | 1, 20 | mpd 13 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 class class class wbr 3929 cr 7622 clt 7803 cle 7804 cz 9057 |
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 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-dc 820 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-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-inn 8724 df-n0 8981 df-z 9058 |
This theorem is referenced by: uzin 9361 exfzdc 10020 modfzo0difsn 10171 fzfig 10206 iseqf1olemjpcl 10271 iseqf1olemqpcl 10272 seq3f1oleml 10279 seq3f1o 10280 fser0const 10292 uzin2 10762 2zsupmax 11000 sumeq2 11131 summodclem2a 11153 fsum3 11159 fsumcl2lem 11170 fsumadd 11178 sumsnf 11181 fsummulc2 11220 explecnv 11277 prodeq2 11329 prodmodclem3 11347 prodmodclem2a 11348 infssuzex 11645 |
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