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Mirrors > Home > ILE Home > Th. List > zsubcl | Unicode version |
Description: Closure of subtraction of integers. (Contributed by NM, 11-May-2004.) |
Ref | Expression |
---|---|
zsubcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9188 | . . 3 | |
2 | zcn 9188 | . . 3 | |
3 | negsub 8138 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | znegcl 9214 | . . 3 | |
6 | zaddcl 9223 | . . 3 | |
7 | 5, 6 | sylan2 284 | . 2 |
8 | 4, 7 | eqeltrrd 2242 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 (class class class)co 5837 cc 7743 caddc 7748 cmin 8061 cneg 8062 cz 9183 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-addcom 7845 ax-addass 7847 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-0id 7853 ax-rnegex 7854 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-ltadd 7861 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-id 4266 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-inn 8850 df-n0 9107 df-z 9184 |
This theorem is referenced by: ztri3or 9226 zrevaddcl 9233 znnsub 9234 nzadd 9235 znn0sub 9248 zneo 9284 zsubcld 9310 eluzsubi 9485 fzen 9969 uzsubsubfz 9973 fzrev 10010 fzrev2 10011 fzrevral2 10032 fzshftral 10034 fz0fzdiffz0 10056 difelfzle 10060 difelfznle 10061 elfzomelpfzo 10157 zmodcl 10270 frecfzen2 10353 facndiv 10642 bccmpl 10657 bcpasc 10669 hashfz 10724 moddvds 11729 modmulconst 11753 dvds2sub 11756 dvdssub2 11764 dvdssubr 11768 fzocongeq 11785 odd2np1 11799 omoe 11822 omeo 11824 divalgb 11851 divalgmod 11853 ndvdsadd 11857 nn0seqcvgd 11962 congr 12021 cncongr1 12024 cncongr2 12025 prmdiv 12156 prmdiveq 12157 pythagtriplem4 12189 pythagtriplem8 12193 difsqpwdvds 12258 |
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