Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 9202 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
2 | oddp1d2 11812 | . . 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 9114 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
5 | 1red 7905 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
6 | nn0ge0 9130 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
7 | 0le1 8370 | . . . . . . . . . 10 ⊢ 0 ≤ 1 | |
8 | 7 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 1) |
9 | 4, 5, 6, 8 | addge0d 8411 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ (𝑁 + 1)) |
10 | peano2nn0 9145 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
11 | 10 | nn0red 9159 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℝ) |
12 | 2re 8918 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
13 | 12 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
14 | 2pos 8939 | . . . . . . . . . 10 ⊢ 0 < 2 | |
15 | 14 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 < 2) |
16 | ge0div 8757 | . . . . . . . . 9 ⊢ (((𝑁 + 1) ∈ ℝ ∧ 2 ∈ ℝ ∧ 0 < 2) → (0 ≤ (𝑁 + 1) ↔ 0 ≤ ((𝑁 + 1) / 2))) | |
17 | 11, 13, 15, 16 | syl3anc 1227 | . . . . . . . 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 9195 | . . . . 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 9202 | . . 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 11830 | . 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 2135 class class class wbr 3976 (class class class)co 5836 ℝcr 7743 0cc0 7744 1c1 7745 + caddc 7747 < clt 7924 ≤ cle 7925 − cmin 8060 / cdiv 8559 2c2 8899 ℕ0cn0 9105 ℤcz 9182 ∥ cdvds 11713 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-xor 1365 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-dvds 11714 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |