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Mirrors > Home > ILE Home > Th. List > nn0oddm1d2 | GIF version |
Description: A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Ref | Expression |
---|---|
nn0oddm1d2 | ⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 9074 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | oddp1d2 11587 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
4 | nn0re 8986 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
5 | 1red 7781 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
6 | nn0ge0 9002 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
7 | 0le1 8243 | . . . . . . . . . 10 ⊢ 0 ≤ 1 | |
8 | 7 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 1) |
9 | 4, 5, 6, 8 | addge0d 8284 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ (𝑁 + 1)) |
10 | peano2nn0 9017 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
11 | 10 | nn0red 9031 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℝ) |
12 | 2re 8790 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
14 | 2pos 8811 | . . . . . . . . . 10 ⊢ 0 < 2 | |
15 | 14 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 < 2) |
16 | ge0div 8629 | . . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 + 1) ↔ 0 ≤ ((𝑁 + 1) / 2))) | |
17 | 11, 13, 15, 16 | syl3anc 1216 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (0 ≤ (𝑁 + 1) ↔ 0 ≤ ((𝑁 + 1) / 2))) |
18 | 9, 17 | mpbid 146 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ ((𝑁 + 1) / 2)) |
19 | 18 | anim1i 338 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (0 ≤ ((𝑁 + 1) / 2) ∧ ((𝑁 + 1) / 2) ∈ ℤ)) |
20 | 19 | ancomd 265 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 + 1) / 2))) |
21 | elnn0z 9067 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ0 ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 ≤ ((𝑁 + 1) / 2))) | |
22 | 20, 21 | sylibr 133 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ0) |
23 | 22 | ex 114 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ0)) |
24 | nn0z 9074 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ0 → ((𝑁 + 1) / 2) ∈ ℤ) | |
25 | 23, 24 | impbid1 141 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℕ0)) |
26 | nn0ob 11605 | . 2 ⊢ (𝑁 ∈ ℕ0 → (((𝑁 + 1) / 2) ∈ ℕ0 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) | |
27 | 3, 25, 26 | 3bitrd 213 | 1 ⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ 𝑁 ↔ ((𝑁 − 1) / 2) ∈ ℕ0)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 ≤ cle 7801 − cmin 7933 / cdiv 8432 2c2 8771 ℕ0cn0 8977 ℤcz 9054 ∥ cdvds 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-xor 1354 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-dvds 11494 |
This theorem is referenced by: (None) |
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