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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 9144 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | elnn0 9137 | . . . . . . 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 9214 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
10 | 1, 8, 9 | sylanbrc 415 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) |
11 | nn0ge0 9160 | . . 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 9150 | . . . . . 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 9142 | . . . . 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 7921 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 0 ∈ ℝ) | |
23 | zre 9216 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
24 | 23 | adantr 274 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → 𝑁 ∈ ℝ) |
25 | 22, 24 | lenltd 8037 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → (0 ≤ 𝑁 ↔ ¬ 𝑁 < 0)) |
26 | 21, 25 | mpbid 146 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 0 ≤ 𝑁) → ¬ 𝑁 < 0) |
27 | nngt0 8903 | . . . . . . 7 ⊢ (-𝑁 ∈ ℕ → 0 < -𝑁) | |
28 | 24 | lt0neg1d 8434 | . . . . . . 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 3989 ℝcr 7773 0cc0 7774 < clt 7954 ≤ cle 7955 -cneg 8091 ℕcn 8878 ℕ0cn0 9135 ℤcz 9212 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 |
This theorem is referenced by: nn0zrab 9237 znn0sub 9277 nn0ind 9326 fnn0ind 9328 fznn0 10069 elfz0ubfz0 10081 elfz0fzfz0 10082 fz0fzelfz0 10083 elfzmlbp 10088 difelfzle 10090 difelfznle 10091 elfzo0z 10140 fzofzim 10144 ubmelm1fzo 10182 flqge0nn0 10249 zmodcl 10300 modqmuladdnn0 10324 modsumfzodifsn 10352 uzennn 10392 zsqcl2 10553 nn0abscl 11049 geolim2 11475 cvgratnnlemabsle 11490 oexpneg 11836 oddnn02np1 11839 evennn02n 11841 nn0ehalf 11862 nn0oddm1d2 11868 divalgb 11884 dfgcd2 11969 uzwodc 11992 algcvga 12005 hashgcdlem 12192 pockthlem 12308 ennnfoneleminc 12366 |
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