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Mirrors > Home > ILE Home > Th. List > elznn0 | GIF version |
Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
elznn0 | ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9184 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | elnn0 9107 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
4 | elnn0 9107 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ0 ↔ (-𝑁 ∈ ℕ ∨ -𝑁 = 0)) | |
5 | recn 7877 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ) | |
6 | 0cn 7882 | . . . . . . . . 9 ⊢ 0 ∈ ℂ | |
7 | negcon1 8141 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 0 ∈ ℂ) → (-𝑁 = 0 ↔ -0 = 𝑁)) | |
8 | 5, 6, 7 | sylancl 410 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ -0 = 𝑁)) |
9 | neg0 8135 | . . . . . . . . . 10 ⊢ -0 = 0 | |
10 | 9 | eqeq1i 2172 | . . . . . . . . 9 ⊢ (-0 = 𝑁 ↔ 0 = 𝑁) |
11 | eqcom 2166 | . . . . . . . . 9 ⊢ (0 = 𝑁 ↔ 𝑁 = 0) | |
12 | 10, 11 | bitri 183 | . . . . . . . 8 ⊢ (-0 = 𝑁 ↔ 𝑁 = 0) |
13 | 8, 12 | bitrdi 195 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ 𝑁 = 0)) |
14 | 13 | orbi2d 780 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 ∈ ℕ ∨ -𝑁 = 0) ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
15 | 4, 14 | syl5bb 191 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (-𝑁 ∈ ℕ0 ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
16 | 3, 15 | orbi12d 783 | . . . 4 ⊢ (𝑁 ∈ ℝ → ((𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))) |
17 | 3orass 970 | . . . . 5 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
18 | orcom 718 | . . . . 5 ⊢ ((𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0)) | |
19 | orordir 764 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) | |
20 | 17, 18, 19 | 3bitrri 206 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
21 | 16, 20 | bitr2di 196 | . . 3 ⊢ (𝑁 ∈ ℝ → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
22 | 21 | pm5.32i 450 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
23 | 1, 22 | bitri 183 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∨ wo 698 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 ℂcc 7742 ℝcr 7743 0cc0 7744 -cneg 8061 ℕcn 8848 ℕ0cn0 9105 ℤcz 9182 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 df-n0 9106 df-z 9183 |
This theorem is referenced by: peano2z 9218 zmulcl 9235 elz2 9253 expnegzap 10479 expaddzaplem 10488 odd2np1 11795 bezoutlemzz 11920 bezoutlemaz 11921 bezoutlembz 11922 |
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