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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 47003 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 480 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | elnn0 12512 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
4 | nnm1ge0 12668 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | |
5 | nnre 12257 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | peano2rem 11565 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℝ) |
8 | 2re 12324 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
10 | 2pos 12353 | . . . . . . . . 9 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
12 | ge0div 12119 | . . . . . . . 8 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) | |
13 | 7, 9, 11, 12 | syl3anc 1368 | . . . . . . 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 2817 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
17 | 0noddALTV 47058 | . . . . . . 7 ⊢ 0 ∉ Odd | |
18 | df-nel 3044 | . . . . . . . 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 | biimtrdi 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 405 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → 0 ≤ ((𝑁 − 1) / 2)) |
26 | elnn0z 12609 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ0 ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 − 1) / 2))) | |
27 | 2, 25, 26 | sylanbrc 581 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ∉ wnel 3043 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ0cn0 12510 ℤcz 12596 Odd codd 46994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-even 46995 df-odd 46996 |
This theorem is referenced by: nn0onn0exALTV 47068 |
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