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Mirrors > Home > MPE Home > Th. List > zeo2 | Structured version Visualization version 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 11667 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | peano2cn 10496 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℂ) |
4 | 2cnd 11387 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
5 | 2ne0 11420 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
7 | 3, 4, 6 | divcan2d 11093 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = (𝑁 + 1)) |
8 | 1, 4, 6 | divcan2d 11093 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · (𝑁 / 2)) = 𝑁) |
9 | 8 | oveq1d 6891 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · (𝑁 / 2)) + 1) = (𝑁 + 1)) |
10 | 7, 9 | eqtr4d 2834 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1)) |
11 | zneo 11746 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1)) | |
12 | 11 | expcom 403 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1))) |
13 | 12 | necon2bd 2985 | . . 3 ⊢ ((𝑁 / 2) ∈ ℤ → ((2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1) → ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
14 | 10, 13 | syl5com 31 | . 2 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ → ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
15 | zeo 11749 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | |
16 | 15 | ord 891 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ (𝑁 / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℤ)) |
17 | 16 | con1d 142 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ ((𝑁 + 1) / 2) ∈ ℤ → (𝑁 / 2) ∈ ℤ)) |
18 | 14, 17 | impbid 204 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 (class class class)co 6876 ℂcc 10220 0cc0 10222 1c1 10223 + caddc 10225 · cmul 10227 / cdiv 10974 2c2 11364 ℤcz 11662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-n0 11577 df-z 11663 |
This theorem is referenced by: zesq 13237 oddfl 40223 evennodd 42326 oddneven 42327 dignn0flhalflem1 43196 |
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