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Mirrors > Home > ILE Home > Th. List > elnn0z | GIF version |
Description: Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elnn0z | ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 9202 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | elnn0 9195 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
3 | 2 | biimpi 120 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
4 | 3 | orcomd 730 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) |
5 | 3mix1 1167 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) | |
6 | 3mix2 1168 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) | |
7 | 5, 6 | jaoi 717 | . . . . 5 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ) → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
8 | 4, 7 | syl 14 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
9 | elz 9272 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
10 | 1, 8, 9 | sylanbrc 417 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
11 | nn0ge0 9218 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
12 | 10, 11 | jca 306 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
13 | 9 | simprbi 275 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
14 | 13 | adantr 276 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
15 | 0nn0 9208 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | eleq1 2251 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ0 ↔ 0 ∈ ℕ0)) | |
17 | 15, 16 | mpbiri 168 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 ∈ ℕ0) |
18 | 17 | a1i 9 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 = 0 → 𝑁 ∈ ℕ0)) |
19 | nnnn0 9200 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
20 | 19 | a1i 9 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)) |
21 | simpr 110 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 0 ≤ 𝑁) | |
22 | 0red 7975 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 0 ∈ ℝ) | |
23 | zre 9274 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
24 | 23 | adantr 276 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝑁 ∈ ℝ) |
25 | 22, 24 | lenltd 8092 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (0 ≤ 𝑁 ↔ ¬ 𝑁 < 0)) |
26 | 21, 25 | mpbid 147 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ¬ 𝑁 < 0) |
27 | nngt0 8961 | . . . . . . 7 ⊢ (-𝑁 ∈ ℕ → 0 < -𝑁) | |
28 | 24 | lt0neg1d 8489 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁)) |
29 | 27, 28 | imbitrrid 156 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (-𝑁 ∈ ℕ → 𝑁 < 0)) |
30 | 26, 29 | mtod 664 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ¬ -𝑁 ∈ ℕ) |
31 | 30 | pm2.21d 620 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (-𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)) |
32 | 18, 20, 31 | 3jaod 1314 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → 𝑁 ∈ ℕ0)) |
33 | 14, 32 | mpd 13 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝑁 ∈ ℕ0) |
34 | 12, 33 | impbii 126 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 ∨ w3o 978 = wceq 1363 ∈ wcel 2159 class class class wbr 4017 ℝcr 7827 0cc0 7828 < clt 8009 ≤ cle 8010 -cneg 8146 ℕcn 8936 ℕ0cn0 9193 ℤcz 9270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-iota 5192 df-fun 5232 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-inn 8937 df-n0 9194 df-z 9271 |
This theorem is referenced by: nn0zrab 9295 znn0sub 9335 nn0ind 9384 fnn0ind 9386 fznn0 10130 elfz0ubfz0 10142 elfz0fzfz0 10143 fz0fzelfz0 10144 elfzmlbp 10149 difelfzle 10151 difelfznle 10152 elfzo0z 10201 fzofzim 10205 ubmelm1fzo 10243 flqge0nn0 10310 zmodcl 10361 modqmuladdnn0 10385 modsumfzodifsn 10413 uzennn 10453 zsqcl2 10615 nn0abscl 11111 geolim2 11537 cvgratnnlemabsle 11552 oexpneg 11899 oddnn02np1 11902 evennn02n 11904 nn0ehalf 11925 nn0oddm1d2 11931 divalgb 11947 dfgcd2 12032 uzwodc 12055 algcvga 12068 hashgcdlem 12255 pockthlem 12371 ennnfoneleminc 12429 |
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