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Mirrors > Home > MPE Home > Th. List > nnoddm1d2 | Structured version Visualization version GIF version |
Description: A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Ref | Expression |
---|---|
nnoddm1d2 | ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 11751 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | oddp1d2 15486 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
4 | peano2nn 11388 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | nnred 11391 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ) |
6 | 2re 11449 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
8 | nnre 11382 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 10377 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
10 | nngt0 11407 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
11 | 0lt1 10897 | . . . . . . . . 9 ⊢ 0 < 1 | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1) |
13 | 8, 9, 10, 12 | addgt0d 10950 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1)) |
14 | 2pos 11485 | . . . . . . . 8 ⊢ 0 < 2 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
16 | 5, 7, 13, 15 | divgt0d 11313 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 < ((𝑁 + 1) / 2)) |
17 | 16 | anim1ci 609 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) |
18 | elnnz 11738 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) | |
19 | 17, 18 | sylibr 226 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ) |
20 | 19 | ex 403 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ)) |
21 | nnz 11751 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
22 | 20, 21 | impbid1 217 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
23 | 3, 22 | bitrd 271 | 1 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2106 class class class wbr 4886 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 < clt 10411 / cdiv 11032 ℕcn 11374 2c2 11430 ℤcz 11728 ∥ cdvds 15387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-dvds 15388 |
This theorem is referenced by: gausslemma2dlem0b 25534 |
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