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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 9133 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | elnn0 9126 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
3 | 2 | biimpi 119 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
4 | 3 | orcomd 724 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ)) |
5 | 3mix1 1161 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) | |
6 | 3mix2 1162 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) | |
7 | 5, 6 | jaoi 711 | . . . . 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 9203 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
10 | 1, 8, 9 | sylanbrc 415 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
11 | nn0ge0 9149 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
12 | 10, 11 | jca 304 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
13 | 9 | simprbi 273 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
14 | 13 | adantr 274 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
15 | 0nn0 9139 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
16 | eleq1 2233 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ0 ↔ 0 ∈ ℕ0)) | |
17 | 15, 16 | mpbiri 167 | . . . . 5 ⊢ (𝑁 = 0 → 𝑁 ∈ ℕ0) |
18 | 17 | a1i 9 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 = 0 → 𝑁 ∈ ℕ0)) |
19 | nnnn0 9131 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
20 | 19 | a1i 9 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)) |
21 | simpr 109 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 0 ≤ 𝑁) | |
22 | 0red 7910 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 0 ∈ ℝ) | |
23 | zre 9205 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
24 | 23 | adantr 274 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝑁 ∈ ℝ) |
25 | 22, 24 | lenltd 8026 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (0 ≤ 𝑁 ↔ ¬ 𝑁 < 0)) |
26 | 21, 25 | mpbid 146 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ¬ 𝑁 < 0) |
27 | nngt0 8892 | . . . . . . 7 ⊢ (-𝑁 ∈ ℕ → 0 < -𝑁) | |
28 | 24 | lt0neg1d 8423 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁)) |
29 | 27, 28 | syl5ibr 155 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (-𝑁 ∈ ℕ → 𝑁 < 0)) |
30 | 26, 29 | mtod 658 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ¬ -𝑁 ∈ ℕ) |
31 | 30 | pm2.21d 614 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (-𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)) |
32 | 18, 20, 31 | 3jaod 1299 | . . 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 125 | 1 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℤ ∧ 0 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ℝcr 7762 0cc0 7763 < clt 7943 ≤ cle 7944 -cneg 8080 ℕcn 8867 ℕ0cn0 9124 ℤcz 9201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-ltadd 7879 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-inn 8868 df-n0 9125 df-z 9202 |
This theorem is referenced by: nn0zrab 9226 znn0sub 9266 nn0ind 9315 fnn0ind 9317 fznn0 10058 elfz0ubfz0 10070 elfz0fzfz0 10071 fz0fzelfz0 10072 elfzmlbp 10077 difelfzle 10079 difelfznle 10080 elfzo0z 10129 fzofzim 10133 ubmelm1fzo 10171 flqge0nn0 10238 zmodcl 10289 modqmuladdnn0 10313 modsumfzodifsn 10341 uzennn 10381 zsqcl2 10542 nn0abscl 11038 geolim2 11464 cvgratnnlemabsle 11479 oexpneg 11825 oddnn02np1 11828 evennn02n 11830 nn0ehalf 11851 nn0oddm1d2 11857 divalgb 11873 dfgcd2 11958 uzwodc 11981 algcvga 11994 hashgcdlem 12181 pockthlem 12297 ennnfoneleminc 12355 |
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