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Mirrors > Home > ILE Home > Th. List > nnnegz | GIF version |
Description: The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.) |
Ref | Expression |
---|---|
nnnegz | ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnre 7921 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
2 | 1 | renegcld 7378 | . 2 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℝ) |
3 | nncn 7922 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
4 | negneg 7261 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → --𝑁 = 𝑁) | |
5 | 4 | eleq1d 2106 | . . . . 5 ⊢ (𝑁 ∈ ℂ → (--𝑁 ∈ ℕ ↔ 𝑁 ∈ ℕ)) |
6 | 5 | biimprd 147 | . . . 4 ⊢ (𝑁 ∈ ℂ → (𝑁 ∈ ℕ → --𝑁 ∈ ℕ)) |
7 | 3, 6 | mpcom 32 | . . 3 ⊢ (𝑁 ∈ ℕ → --𝑁 ∈ ℕ) |
8 | 7 | 3mix3d 1081 | . 2 ⊢ (𝑁 ∈ ℕ → (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ)) |
9 | elz 8247 | . 2 ⊢ (-𝑁 ∈ ℤ ↔ (-𝑁 ∈ ℝ ∧ (-𝑁 = 0 ∨ -𝑁 ∈ ℕ ∨ --𝑁 ∈ ℕ))) | |
10 | 2, 8, 9 | sylanbrc 394 | 1 ⊢ (𝑁 ∈ ℕ → -𝑁 ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 ℂcc 6887 ℝcr 6888 0cc0 6889 -cneg 7183 ℕcn 7914 ℤcz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 df-inn 7915 df-z 8246 |
This theorem is referenced by: znegcl 8276 neg1z 8277 zeo 8343 btwnz 8357 expaddzaplem 9298 |
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