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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 46237 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 483 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | elnn0 12470 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
4 | nnm1ge0 12626 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | |
5 | nnre 12215 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | peano2rem 11523 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℝ) |
8 | 2re 12282 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
10 | 2pos 12311 | . . . . . . . . 9 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
12 | ge0div 12077 | . . . . . . . 8 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) | |
13 | 7, 9, 11, 12 | syl3anc 1372 | . . . . . . 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 2822 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
17 | 0noddALTV 46292 | . . . . . . 7 ⊢ 0 ∉ Odd | |
18 | df-nel 3048 | . . . . . . . 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 253 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
23 | 15, 22 | jaoi 856 | . . . 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 408 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → 0 ≤ ((𝑁 − 1) / 2)) |
26 | elnn0z 12567 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ0 ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 − 1) / 2))) | |
27 | 2, 25, 26 | sylanbrc 584 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ∉ wnel 3047 class class class wbr 5147 (class class class)co 7404 ℝcr 11105 0cc0 11106 1c1 11107 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ0cn0 12468 ℤcz 12554 Odd codd 46228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-even 46229 df-odd 46230 |
This theorem is referenced by: nn0onn0exALTV 46302 |
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