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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 12459 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | negeq 11351 | . . . . 5 ⊢ (𝑁 = 0 → -𝑁 = -0) | |
3 | neg0 11405 | . . . . 5 ⊢ -0 = 0 | |
4 | 2, 3 | eqtrdi 2793 | . . . 4 ⊢ (𝑁 = 0 → -𝑁 = 0) |
5 | 0z 12468 | . . . 4 ⊢ 0 ∈ ℤ | |
6 | 4, 5 | eqeltrdi 2846 | . . 3 ⊢ (𝑁 = 0 → -𝑁 ∈ ℤ) |
7 | nnnegz 12460 | . . 3 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
8 | nnz 12478 | . . 3 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
9 | 6, 7, 8 | 3jaoi 1427 | . 2 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → -𝑁 ∈ ℤ) |
10 | 1, 9 | simplbiim 505 | 1 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ℝcr 11008 0cc0 11009 -cneg 11344 ℕcn 12111 ℤcz 12457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-sub 11345 df-neg 11346 df-nn 12112 df-z 12458 |
This theorem is referenced by: znegclb 12498 nn0negz 12499 zsubcl 12503 zeo 12547 zindd 12562 znegcld 12567 zriotaneg 12574 uzneg 12741 zmax 12824 rebtwnz 12826 qnegcl 12845 fzsubel 13431 fzosubel 13585 ceilid 13710 modcyc2 13766 expsub 13970 seqshft 14930 climshft 15418 negdvdsb 16115 dvdsnegb 16116 summodnegmod 16129 dvdssub 16146 odd2np1 16183 divalglem6 16240 bitscmp 16278 gcdneg 16362 neggcd 16363 gcdaddmlem 16364 gcdabsOLD 16372 lcmneg 16439 neglcm 16440 lcmabs 16441 mulgaddcomlem 18858 mulgneg2 18869 mulgsubdir 18875 cycsubgcl 18958 zaddablx 19609 cyggeninv 19619 zsubrg 20803 zringmulg 20830 zringinvg 20839 aaliou3lem9 25662 sinperlem 25789 wilthlem3 26371 basellem3 26384 basellem4 26385 basellem8 26389 basellem9 26390 lgsneg 26621 lgsdir2lem4 26628 lgsdir2lem5 26629 ex-fl 29220 ex-mod 29222 pell1234qrdich 41093 rmxyneg 41153 monotoddzzfi 41175 monotoddzz 41176 oddcomabszz 41177 jm2.24 41196 acongtr 41211 fzneg 41215 jm2.26a 41233 cosknegpi 44011 enege 45738 onego 45739 0nodd 46005 2zrngagrp 46142 zlmodzxzequap 46481 flsubz 46504 digvalnn0 46586 dig0 46593 dig2nn0 46598 |
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