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Mirrors > Home > ILE Home > Th. List > zdceq | Unicode version |
Description: Equality of integers is decidable. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zdceq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ztri3or 9211 | . 2 | |
2 | zre 9172 | . . . 4 | |
3 | ltne 7963 | . . . . . . . 8 | |
4 | 3 | necomd 2413 | . . . . . . 7 |
5 | olc 701 | . . . . . . . 8 | |
6 | dcne 2338 | . . . . . . . 8 DECID | |
7 | 5, 6 | sylibr 133 | . . . . . . 7 DECID |
8 | 4, 7 | syl 14 | . . . . . 6 DECID |
9 | 8 | ex 114 | . . . . 5 DECID |
10 | 9 | adantr 274 | . . . 4 DECID |
11 | 2, 10 | sylan 281 | . . 3 DECID |
12 | orc 702 | . . . . 5 | |
13 | 12, 6 | sylibr 133 | . . . 4 DECID |
14 | 13 | a1i 9 | . . 3 DECID |
15 | zre 9172 | . . . . 5 | |
16 | ltne 7963 | . . . . . . 7 | |
17 | 16, 7 | syl 14 | . . . . . 6 DECID |
18 | 17 | ex 114 | . . . . 5 DECID |
19 | 15, 18 | syl 14 | . . . 4 DECID |
20 | 19 | adantl 275 | . . 3 DECID |
21 | 11, 14, 20 | 3jaod 1286 | . 2 DECID |
22 | 1, 21 | mpd 13 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 DECID wdc 820 w3o 962 wceq 1335 wcel 2128 wne 2327 class class class wbr 3966 cr 7732 clt 7913 cz 9168 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 |
This theorem is referenced by: nn0n0n1ge2b 9244 nn0lt2 9246 prime 9264 elnn1uz2 9519 iseqf1olemqcl 10389 iseqf1olemnab 10391 iseqf1olemab 10392 seq3f1olemstep 10404 exp3val 10425 hashfzp1 10702 fprod1p 11500 dvdsdc 11698 zdvdsdc 11712 dvdsabseq 11743 alzdvds 11750 fzo0dvdseq 11753 gcdmndc 11835 gcdsupex 11845 gcdsupcl 11846 gcd0id 11867 gcdaddm 11872 dfgcd2 11902 gcdmultiplez 11909 dvdssq 11919 nn0seqcvgd 11922 algcvgblem 11930 eucalgval2 11934 lcmmndc 11943 lcmdvds 11960 lcmid 11961 mulgcddvds 11975 cncongr2 11985 isprm3 11999 isprm4 12000 prm2orodd 12007 rpexp 12032 phivalfi 12091 phiprmpw 12101 phimullem 12104 eulerthlemfi 12107 hashgcdeq 12118 phisum 12119 ennnfonelemim 12195 unbendc 12227 strsetsid 12265 nninffeq 13634 nconstwlpolem 13677 |
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