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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 11382 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | peano2cn 10208 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑁 + 1) ∈ ℂ) |
4 | 2cnd 11093 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
5 | 2ne0 11113 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ≠ 0) |
7 | 3, 4, 6 | divcan2d 10803 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = (𝑁 + 1)) |
8 | 1, 4, 6 | divcan2d 10803 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (2 · (𝑁 / 2)) = 𝑁) |
9 | 8 | oveq1d 6665 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((2 · (𝑁 / 2)) + 1) = (𝑁 + 1)) |
10 | 7, 9 | eqtr4d 2659 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 · ((𝑁 + 1) / 2)) = ((2 · (𝑁 / 2)) + 1)) |
11 | zneo 11460 | . . . . 5 ⊢ ((((𝑁 + 1) / 2) ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1)) | |
12 | 11 | expcom 451 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℤ → (((𝑁 + 1) / 2) ∈ ℤ → (2 · ((𝑁 + 1) / 2)) ≠ ((2 · (𝑁 / 2)) + 1))) |
13 | 12 | necon2bd 2810 | . . 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 11463 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | |
16 | 15 | ord 392 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ (𝑁 / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℤ)) |
17 | 16 | con1d 139 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ ((𝑁 + 1) / 2) ∈ ℤ → (𝑁 / 2) ∈ ℤ)) |
18 | 14, 17 | impbid 202 | 1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ¬ ((𝑁 + 1) / 2) ∈ ℤ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 / cdiv 10684 2c2 11070 ℤcz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 |
This theorem is referenced by: zesq 12987 oddfl 39489 evennodd 41556 oddneven 41557 dignn0flhalflem1 42409 |
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