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Mirrors > Home > ILE Home > Th. List > znegcl | 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 8353 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | negeq 7301 | . . . . . 6 ⊢ (𝑁 = 0 → -𝑁 = -0) | |
3 | neg0 7354 | . . . . . 6 ⊢ -0 = 0 | |
4 | 2, 3 | syl6eq 2129 | . . . . 5 ⊢ (𝑁 = 0 → -𝑁 = 0) |
5 | 0z 8362 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | 4, 5 | syl6eqel 2169 | . . . 4 ⊢ (𝑁 = 0 → -𝑁 ∈ ℤ) |
7 | nnnegz 8354 | . . . 4 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
8 | nnz 8370 | . . . 4 ⊢ (-𝑁 ∈ ℕ → -𝑁 ∈ ℤ) | |
9 | 6, 7, 8 | 3jaoi 1234 | . . 3 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → -𝑁 ∈ ℤ) |
10 | 9 | adantl 271 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → -𝑁 ∈ ℤ) |
11 | 1, 10 | sylbi 119 | 1 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∨ w3o 918 = wceq 1284 ∈ wcel 1433 ℝcr 6980 0cc0 6981 -cneg 7280 ℕcn 8039 ℤcz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-z 8352 |
This theorem is referenced by: znegclb 8384 nn0negz 8385 peano2zm 8389 zsubcl 8392 zeo 8452 zindd 8465 znegcld 8471 uzneg 8637 qnegcl 8721 fzsubel 9078 fzosubel 9203 ceilid 9317 modqcyc2 9362 expsubap 9524 climshft 10143 negdvdsb 10211 dvdsnegb 10212 summodnegmod 10226 dvdssub 10240 odd2np1 10272 gcdneg 10373 neggcd 10374 gcdabs 10379 bezoutlemaz 10392 bezoutlembz 10393 lcmneg 10456 neglcm 10457 lcmabs 10458 ex-fl 10563 |
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