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| Mirrors > Home > MPE Home > Th. List > znegcl | Structured version Visualization version GIF version | ||
| Description: Closure law for negative integers. (Contributed by NM, 9-May-2004.) | 
| Ref | Expression | 
|---|---|
| znegcl | ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elz 12617 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
| 2 | negeq 11501 | . . . . 5 ⊢ (𝑁 = 0 → -𝑁 = -0) | |
| 3 | neg0 11556 | . . . . 5 ⊢ -0 = 0 | |
| 4 | 2, 3 | eqtrdi 2792 | . . . 4 ⊢ (𝑁 = 0 → -𝑁 = 0) | 
| 5 | 0z 12626 | . . . 4 ⊢ 0 ∈ ℤ | |
| 6 | 4, 5 | eqeltrdi 2848 | . . 3 ⊢ (𝑁 = 0 → -𝑁 ∈ ℤ) | 
| 7 | nnnegz 12618 | . . 3 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
| 8 | nnz 12636 | . . 3 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
| 9 | 6, 7, 8 | 3jaoi 1429 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → -𝑁 ∈ ℤ) | 
| 10 | 1, 9 | simplbiim 504 | 1 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ w3o 1085 = wceq 1539 ∈ wcel 2107 ℝcr 11155 0cc0 11156 -cneg 11494 ℕcn 12267 ℤcz 12615 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 df-neg 11496 df-nn 12268 df-z 12616 | 
| This theorem is referenced by: znegclb 12656 nn0negz 12657 zsubcl 12661 zeo 12706 zindd 12721 znegcld 12726 zriotaneg 12733 uzneg 12899 zmax 12988 rebtwnz 12990 qnegcl 13009 fzsubel 13601 fzosubel 13764 ceilid 13892 modcyc2 13948 expsub 14152 seqshft 15125 climshft 15613 negdvdsb 16311 dvdsnegb 16312 summodnegmod 16325 dvdssub 16342 odd2np1 16379 divalglem6 16436 bitscmp 16476 gcdneg 16560 neggcd 16561 gcdaddmlem 16562 lcmneg 16641 neglcm 16642 lcmabs 16643 mulgaddcomlem 19116 mulgneg2 19127 mulgsubdir 19133 cycsubgcl 19225 zaddablx 19891 cyggeninv 19902 zsubrg 21439 zringsub 21467 zringmulg 21468 zringinvg 21477 pzriprnglem4 21496 aaliou3lem9 26393 sinperlem 26523 wilthlem3 27114 basellem3 27127 basellem4 27128 basellem8 27132 basellem9 27133 lgsneg 27366 lgsdir2lem4 27373 lgsdir2lem5 27374 ex-fl 30467 ex-mod 30469 pell1234qrdich 42877 rmxyneg 42937 monotoddzzfi 42959 monotoddzz 42960 oddcomabszz 42961 jm2.24 42980 acongtr 42995 fzneg 42999 jm2.26a 43017 cosknegpi 45889 enege 47637 onego 47638 0nodd 48091 2zrngagrp 48170 zlmodzxzequap 48421 flsubz 48444 digvalnn0 48525 dig0 48532 dig2nn0 48537 | 
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