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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 8356 | . . 3 | |
2 | zcn 8356 | . . 3 | |
3 | negsub 7356 | . . 3 | |
4 | 1, 2, 3 | syl2an 283 | . 2 |
5 | znegcl 8382 | . . 3 | |
6 | zaddcl 8391 | . . 3 | |
7 | 5, 6 | sylan2 280 | . 2 |
8 | 4, 7 | eqeltrrd 2156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 caddc 6984 cmin 7279 cneg 7280 cz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: ztri3or 8394 zrevaddcl 8401 znnsub 8402 nzadd 8403 znn0sub 8416 zneo 8448 zsubcld 8474 eluzsubi 8646 fzen 9062 uzsubsubfz 9066 fzrev 9101 fzrev2 9102 fzrevral2 9123 fzshftral 9125 fz0fzdiffz0 9141 difelfzle 9145 difelfznle 9146 elfzomelpfzo 9240 zmodcl 9346 frecfzen2 9420 facndiv 9666 bccmpl 9681 bcpasc 9693 moddvds 10204 modmulconst 10227 dvds2sub 10230 dvdssub2 10237 dvdssubr 10241 fzocongeq 10258 odd2np1 10272 omoe 10296 omeo 10298 divalgb 10325 divalgmod 10327 ndvdsadd 10331 nn0seqcvgd 10423 congr 10482 cncongr1 10485 cncongr2 10486 |
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