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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 9231 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | elnn0 9154 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
4 | elnn0 9154 | . . . . . 6 ⊢ (-𝑁 ∈ ℕ0 ↔ (-𝑁 ∈ ℕ ∨ -𝑁 = 0)) | |
5 | recn 7922 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ) | |
6 | 0cn 7927 | . . . . . . . . 9 ⊢ 0 ∈ ℂ | |
7 | negcon1 8186 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 0 ∈ ℂ) → (-𝑁 = 0 ↔ -0 = 𝑁)) | |
8 | 5, 6, 7 | sylancl 413 | . . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ -0 = 𝑁)) |
9 | neg0 8180 | . . . . . . . . . 10 ⊢ -0 = 0 | |
10 | 9 | eqeq1i 2185 | . . . . . . . . 9 ⊢ (-0 = 𝑁 ↔ 0 = 𝑁) |
11 | eqcom 2179 | . . . . . . . . 9 ⊢ (0 = 𝑁 ↔ 𝑁 = 0) | |
12 | 10, 11 | bitri 184 | . . . . . . . 8 ⊢ (-0 = 𝑁 ↔ 𝑁 = 0) |
13 | 8, 12 | bitrdi 196 | . . . . . . 7 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ 𝑁 = 0)) |
14 | 13 | orbi2d 790 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 ∈ ℕ ∨ -𝑁 = 0) ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
15 | 4, 14 | bitrid 192 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (-𝑁 ∈ ℕ0 ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) |
16 | 3, 15 | orbi12d 793 | . . . 4 ⊢ (𝑁 ∈ ℝ → ((𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))) |
17 | 3orass 981 | . . . . 5 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
18 | orcom 728 | . . . . 5 ⊢ ((𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0)) | |
19 | orordir 774 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) | |
20 | 17, 18, 19 | 3bitrri 207 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
21 | 16, 20 | bitr2di 197 | . . 3 ⊢ (𝑁 ∈ ℝ → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
22 | 21 | pm5.32i 454 | . 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
23 | 1, 22 | bitri 184 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ0 ∨ -𝑁 ∈ ℕ0))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∨ wo 708 ∨ w3o 977 = wceq 1353 ∈ wcel 2148 ℂcc 7787 ℝcr 7788 0cc0 7789 -cneg 8106 ℕcn 8895 ℕ0cn0 9152 ℤcz 9229 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-setind 4532 ax-resscn 7881 ax-1cn 7882 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-sub 8107 df-neg 8108 df-n0 9153 df-z 9230 |
This theorem is referenced by: peano2z 9265 zmulcl 9282 elz2 9300 expnegzap 10527 expaddzaplem 10536 odd2np1 11848 bezoutlemzz 11973 bezoutlemaz 11974 bezoutlembz 11975 mulgz 12886 mulgdirlem 12889 mulgdir 12890 mulgass 12895 |
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