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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 8886 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
2 | peano2cn 8054 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℂ) |
4 | 2cnd 8951 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
5 | 2ap0 8971 | . . . . . 6 ⊢ 2 # 0 | |
6 | 5 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 2 # 0) |
7 | 3, 4, 6 | divcanap2d 8709 | . . . 4 ⊢ (𝑁 ∈ ℕ → (2 · ((𝑁 + 1) / 2)) = (𝑁 + 1)) |
8 | 1, 4, 6 | divcanap2d 8709 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2 · (𝑁 / 2)) = 𝑁) |
9 | 8 | oveq1d 5868 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2 · (𝑁 / 2)) + 1) = (𝑁 + 1)) |
10 | 7, 9 | eqtr4d 2206 | . . 3 ⊢ (𝑁 ∈ ℕ → (2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1)) |
11 | nnz 9231 | . . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
12 | nnz 9231 | . . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ) | |
13 | zneo 9313 | . . . . . 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 2398 | . . 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 9314 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ)) | |
19 | 18 | orcomd 724 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℕ ∨ (𝑁 / 2) ∈ ℕ)) |
20 | 19 | ord 719 | . 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 1348 ∈ wcel 2141 ≠ wne 2340 class class class wbr 3989 (class class class)co 5853 ℂcc 7772 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 # cap 8500 / cdiv 8589 ℕcn 8878 2c2 8929 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-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
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-rmo 2456 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-po 4281 df-iso 4282 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-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 |
This theorem is referenced by: nneoi 9316 |
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