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Mirrors > Home > ILE Home > Th. List > znegcl | Unicode version |
Description: Closure law for negative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
znegcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9253 |
. 2
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2 | negeq 8148 |
. . . . . 6
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3 | neg0 8201 |
. . . . . 6
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4 | 2, 3 | eqtrdi 2226 |
. . . . 5
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5 | 0z 9262 |
. . . . 5
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6 | 4, 5 | eqeltrdi 2268 |
. . . 4
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7 | nnnegz 9254 |
. . . 4
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8 | nnz 9270 |
. . . 4
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9 | 6, 7, 8 | 3jaoi 1303 |
. . 3
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10 | 9 | adantl 277 |
. 2
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11 | 1, 10 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-inn 8918 df-z 9252 |
This theorem is referenced by: znegclb 9284 nn0negz 9285 peano2zm 9289 zsubcl 9292 zeo 9356 zindd 9369 znegcld 9375 uzneg 9544 qnegcl 9634 fzsubel 10057 fzosubel 10191 ceilid 10312 modqcyc2 10357 expsubap 10565 climshft 11307 negdvdsb 11809 dvdsnegb 11810 summodnegmod 11824 dvdssub 11840 odd2np1 11872 gcdneg 11977 neggcd 11978 gcdabs 11983 bezoutlemaz 11998 bezoutlembz 11999 lcmneg 12068 neglcm 12069 lcmabs 12070 mulgval 12940 mulgaddcomlem 12959 mulgneg2 12970 mulgsubdir 12976 zsubrg 13366 zringmulg 13379 zringinvg 13385 sinperlem 14122 lgsneg 14318 lgsdir2lem4 14325 lgsdir2lem5 14326 ex-fl 14359 |
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