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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 9027 | . . 3 | |
2 | zcn 9027 | . . 3 | |
3 | negsub 7978 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | znegcl 9053 | . . 3 | |
6 | zaddcl 9062 | . . 3 | |
7 | 5, 6 | sylan2 284 | . 2 |
8 | 4, 7 | eqeltrrd 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 caddc 7591 cmin 7901 cneg 7902 cz 9022 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: ztri3or 9065 zrevaddcl 9072 znnsub 9073 nzadd 9074 znn0sub 9087 zneo 9120 zsubcld 9146 eluzsubi 9321 fzen 9791 uzsubsubfz 9795 fzrev 9832 fzrev2 9833 fzrevral2 9854 fzshftral 9856 fz0fzdiffz0 9875 difelfzle 9879 difelfznle 9880 elfzomelpfzo 9976 zmodcl 10085 frecfzen2 10168 facndiv 10453 bccmpl 10468 bcpasc 10480 hashfz 10535 moddvds 11429 modmulconst 11452 dvds2sub 11455 dvdssub2 11462 dvdssubr 11466 fzocongeq 11483 odd2np1 11497 omoe 11520 omeo 11522 divalgb 11549 divalgmod 11551 ndvdsadd 11555 nn0seqcvgd 11649 congr 11708 cncongr1 11711 cncongr2 11712 |
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