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Mirrors > Home > ILE Home > Th. List > zeo2 | GIF version |
Description: An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) |
Ref | Expression |
---|---|
zeo2 | ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9287 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | peano2cn 8121 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ) | |
3 | 1, 2 | syl 14 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℂ) |
4 | 2cnd 9021 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
5 | 2ap0 9041 | . . . . . 6 ⊢ 2 # 0 | |
6 | 5 | a1i 9 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 # 0) |
7 | 3, 4, 6 | divcanap2d 8778 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = (𝑁 + 1)) |
8 | 1, 4, 6 | divcanap2d 8778 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · (𝑁 / 2)) = 𝑁) |
9 | 8 | oveq1d 5910 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · (𝑁 / 2)) + 1) = (𝑁 + 1)) |
10 | 7, 9 | eqtr4d 2225 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1)) |
11 | zneo 9383 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1)) | |
12 | 11 | expcom 116 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1))) |
13 | 12 | necon2bd 2418 | . . 3 ⊢ ((𝑁 / 2) ∈ ℤ → ((2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1) → ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
14 | 10, 13 | syl5com 29 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ → ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
15 | zeo 9387 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | |
16 | 15 | orcomd 730 | . . 3 ⊢ (𝑁 ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ ∨ (𝑁 / 2) ∈ ℤ)) |
17 | 16 | ord 725 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ ((𝑁 + 1) / 2) ∈ ℤ → (𝑁 / 2) ∈ ℤ)) |
18 | 14, 17 | impbid 129 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 class class class wbr 4018 (class class class)co 5895 ℂcc 7838 0cc0 7840 1c1 7841 + caddc 7843 · cmul 7845 # cap 8567 / cdiv 8658 2c2 8999 ℤcz 9282 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-2 9007 df-n0 9206 df-z 9283 |
This theorem is referenced by: zesq 10669 zeo3 11904 |
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