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Mirrors > Home > ILE Home > Th. List > znegcld | Unicode version |
Description: Closure law for negative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
zred.1 |
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Ref | Expression |
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znegcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zred.1 |
. 2
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2 | znegcl 8844 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-z 8814 |
This theorem is referenced by: ceilqval 9776 ceiqcl 9777 exp3val 10020 expnegap0 10026 expaddzaplem 10061 seq3shft 10335 nn0abscl 10581 climshft2 10758 fsumshftm 10902 eftlub 11043 zdvdsdc 11158 dvdsadd2b 11184 divalglemex 11263 divalglemeuneg 11264 infssuzex 11286 gcdaddm 11316 modgcd 11323 |
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