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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 47559 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | elnn0 12526 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
4 | nnm1ge0 12684 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | |
5 | nnre 12271 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | peano2rem 11574 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℝ) |
8 | 2re 12338 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
10 | 2pos 12367 | . . . . . . . . 9 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
12 | ge0div 12133 | . . . . . . . 8 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) | |
13 | 7, 9, 11, 12 | syl3anc 1370 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) |
14 | 4, 13 | mpbid 232 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((𝑁 − 1) / 2)) |
15 | 14 | a1d 25 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
16 | eleq1 2827 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
17 | 0noddALTV 47614 | . . . . . . 7 ⊢ 0 ∉ Odd | |
18 | df-nel 3045 | . . . . . . . 8 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
19 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) | |
20 | 18, 19 | sylbi 217 | . . . . . . 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 253 | . . . . 5 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
23 | 15, 22 | jaoi 857 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
24 | 3, 23 | sylbi 217 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
25 | 24 | imp 406 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → 0 ≤ ((𝑁 − 1) / 2)) |
26 | elnn0z 12624 | . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 Odd codd 47550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-even 47551 df-odd 47552 |
This theorem is referenced by: nn0onn0exALTV 47624 |
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