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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 44702 | . . 3 ⊢ (𝑁 ∈ Odd → ((𝑁 − 1) / 2) ∈ ℤ) | |
2 | 1 | adantl 485 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℤ) |
3 | elnn0 12057 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
4 | nnm1ge0 12210 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 ≤ (𝑁 − 1)) | |
5 | nnre 11802 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
6 | peano2rem 11110 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℝ) |
8 | 2re 11869 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
10 | 2pos 11898 | . . . . . . . . 9 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
12 | ge0div 11664 | . . . . . . . 8 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) | |
13 | 7, 9, 11, 12 | syl3anc 1373 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 2))) |
14 | 4, 13 | mpbid 235 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 ≤ ((𝑁 − 1) / 2)) |
15 | 14 | a1d 25 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
16 | eleq1 2818 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 ∈ Odd ↔ 0 ∈ Odd )) | |
17 | 0noddALTV 44757 | . . . . . . 7 ⊢ 0 ∉ Odd | |
18 | df-nel 3037 | . . . . . . . 8 ⊢ (0 ∉ Odd ↔ ¬ 0 ∈ Odd ) | |
19 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ 0 ∈ Odd → (0 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) | |
20 | 18, 19 | sylbi 220 | . . . . . . 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 256 | . . . . 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 220 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ Odd → 0 ≤ ((𝑁 − 1) / 2))) |
25 | 24 | imp 410 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → 0 ≤ ((𝑁 − 1) / 2)) |
26 | elnn0z 12154 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ0 ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 − 1) / 2))) | |
27 | 2, 25, 26 | sylanbrc 586 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ∉ wnel 3036 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 < clt 10832 ≤ cle 10833 − cmin 11027 / cdiv 11454 ℕcn 11795 2c2 11850 ℕ0cn0 12055 ℤcz 12141 Odd codd 44693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-even 44694 df-odd 44695 |
This theorem is referenced by: nn0onn0exALTV 44767 |
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