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Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0oALTV | Structured version Visualization version GIF version |
Description: An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.) |
Ref | Expression |
---|---|
nn0oALTV | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddm1div2z 46292 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | elnn0 12473 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
4 | nnm1ge0 12629 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | |
5 | nnre 12218 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | peano2rem 11526 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℝ) |
8 | 2re 12285 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
10 | 2pos 12314 | . . . . . . . . 9 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
12 | ge0div 12080 | . . . . . . . 8 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) | |
13 | 7, 9, 11, 12 | syl3anc 1371 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) |
14 | 4, 13 | mpbid 231 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((𝑁 − 1) / 2)) |
15 | 14 | a1d 25 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
16 | eleq1 2821 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
17 | 0noddALTV 46347 | . . . . . . 7 ⊢ 0 ∉ Odd | |
18 | df-nel 3047 | . . . . . . . 8 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
19 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) | |
20 | 18, 19 | sylbi 216 | . . . . . . 7 ⊢ (0 ∉ Odd → (0 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
21 | 17, 20 | ax-mp 5 | . . . . . 6 ⊢ (0 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2)) |
22 | 16, 21 | syl6bi 252 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
23 | 15, 22 | jaoi 855 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
24 | 3, 23 | sylbi 216 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
25 | 24 | imp 407 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → 0 ≤ ((𝑁 − 1) / 2)) |
26 | elnn0z 12570 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ0 ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 − 1) / 2))) | |
27 | 2, 25, 26 | sylanbrc 583 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 class class class wbr 5148 (class class class)co 7408 ℝcr 11108 0cc0 11109 1c1 11110 < clt 11247 ≤ cle 11248 − cmin 11443 / cdiv 11870 ℕcn 12211 2c2 12266 ℕ0cn0 12471 ℤcz 12557 Odd codd 46283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-even 46284 df-odd 46285 |
This theorem is referenced by: nn0onn0exALTV 46357 |
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