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Mirrors > Home > ILE Home > Th. List > nneo | GIF version |
Description: A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.) |
Ref | Expression |
---|---|
nneo | ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nncn 8728 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | peano2cn 7897 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ) |
4 | 2cnd 8793 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
5 | 2ap0 8813 | . . . . . 6 ⊢ 2 # 0 | |
6 | 5 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 # 0) |
7 | 3, 4, 6 | divcanap2d 8552 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 · ((𝑁 + 1) / 2)) = (𝑁 + 1)) |
8 | 1, 4, 6 | divcanap2d 8552 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2 · (𝑁 / 2)) = 𝑁) |
9 | 8 | oveq1d 5789 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 · (𝑁 / 2)) + 1) = (𝑁 + 1)) |
10 | 7, 9 | eqtr4d 2175 | . . 3 ⊢ (𝑁 ∈ ℕ → (2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1)) |
11 | nnz 9073 | . . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
12 | nnz 9073 | . . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ) | |
13 | zneo 9152 | . . . . . 6 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1)) | |
14 | 11, 12, 13 | syl2an 287 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1)) |
15 | 14 | expcom 115 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℕ → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1))) |
16 | 15 | necon2bd 2366 | . . 3 ⊢ ((𝑁 / 2) ∈ ℕ → ((2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1) → ¬ ((𝑁 + 1) / 2) ∈ ℕ)) |
17 | 10, 16 | syl5com 29 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ → ¬ ((𝑁 + 1) / 2) ∈ ℕ)) |
18 | nneoor 9153 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | |
19 | 18 | orcomd 718 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℕ ∨ (𝑁 / 2) ∈ ℕ)) |
20 | 19 | ord 713 | . 2 ⊢ (𝑁 ∈ ℕ → (¬ ((𝑁 + 1) / 2) ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
21 | 17, 20 | impbid 128 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 # cap 8343 / cdiv 8432 ℕcn 8720 2c2 8771 ℤcz 9054 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-n0 8978 df-z 9055 |
This theorem is referenced by: nneoi 9155 |
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