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Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0eo | Structured version Visualization version GIF version |
Description: A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.) |
Ref | Expression |
---|---|
nn0eo | ⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 11584 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | zeo 11647 | . . 3 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)) |
4 | simpr 479 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℤ) → (𝑁 / 2) ∈ ℤ) | |
5 | nn0re 11485 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
6 | nn0ge0 11502 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
7 | 2re 11274 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
9 | 2pos 11296 | . . . . . . . 8 ⊢ 0 < 2 | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 < 2) |
11 | divge0 11076 | . . . . . . 7 ⊢ (((𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ (𝑁 / 2)) | |
12 | 5, 6, 8, 10, 11 | syl22anc 1474 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ (𝑁 / 2)) |
13 | 12 | adantr 472 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℤ) → 0 ≤ (𝑁 / 2)) |
14 | elnn0z 11574 | . . . . 5 ⊢ ((𝑁 / 2) ∈ ℕ0 ↔ ((𝑁 / 2) ∈ ℤ ∧ 0 ≤ (𝑁 / 2))) | |
15 | 4, 13, 14 | sylanbrc 701 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℤ) → (𝑁 / 2) ∈ ℕ0) |
16 | 15 | ex 449 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℤ → (𝑁 / 2) ∈ ℕ0)) |
17 | simpr 479 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℤ) | |
18 | peano2nn0 11517 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
19 | 18 | nn0red 11536 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℝ) |
20 | 1red 10239 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
21 | 0le1 10735 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
22 | 21 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 1) |
23 | 5, 20, 6, 22 | addge0d 10787 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ (𝑁 + 1)) |
24 | divge0 11076 | . . . . . . 7 ⊢ ((((𝑁 + 1) ∈ ℝ ∧ 0 ≤ (𝑁 + 1)) ∧ (2 ∈ ℝ ∧ 0 < 2)) → 0 ≤ ((𝑁 + 1) / 2)) | |
25 | 19, 23, 8, 10, 24 | syl22anc 1474 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ ((𝑁 + 1) / 2)) |
26 | 25 | adantr 472 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → 0 ≤ ((𝑁 + 1) / 2)) |
27 | elnn0z 11574 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ0 ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 + 1) / 2))) | |
28 | 17, 26, 27 | sylanbrc 701 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ0) |
29 | 28 | ex 449 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ0)) |
30 | 16, 29 | orim12d 919 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0))) |
31 | 3, 30 | mpd 15 | 1 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 2131 class class class wbr 4796 (class class class)co 6805 ℝcr 10119 0cc0 10120 1c1 10121 + caddc 10123 < clt 10258 ≤ cle 10259 / cdiv 10868 2c2 11254 ℕ0cn0 11476 ℤcz 11561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-n0 11477 df-z 11562 |
This theorem is referenced by: flnn0div2ge 42829 dignn0flhalf 42914 |
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