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Mirrors > Home > ILE Home > Th. List > nn0o | GIF version |
Description: An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
Ref | Expression |
---|---|
nn0o | ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0o1gt2 11529 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (𝑁 = 1 ∨ 2 < 𝑁)) | |
2 | 1m1e0 8757 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
3 | 2 | oveq1i 5752 | . . . . . . 7 ⊢ ((1 − 1) / 2) = (0 / 2) |
4 | 2cn 8759 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
5 | 2ap0 8781 | . . . . . . . 8 ⊢ 2 # 0 | |
6 | 4, 5 | div0api 8474 | . . . . . . 7 ⊢ (0 / 2) = 0 |
7 | 3, 6 | eqtri 2138 | . . . . . 6 ⊢ ((1 − 1) / 2) = 0 |
8 | 0nn0 8960 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
9 | 7, 8 | eqeltri 2190 | . . . . 5 ⊢ ((1 − 1) / 2) ∈ ℕ0 |
10 | oveq1 5749 | . . . . . . . 8 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
11 | 10 | oveq1d 5757 | . . . . . . 7 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = ((1 − 1) / 2)) |
12 | 11 | eleq1d 2186 | . . . . . 6 ⊢ (𝑁 = 1 → (((𝑁 − 1) / 2) ∈ ℕ0 ↔ ((1 − 1) / 2) ∈ ℕ0)) |
13 | 12 | adantr 274 | . . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → (((𝑁 − 1) / 2) ∈ ℕ0 ↔ ((1 − 1) / 2) ∈ ℕ0)) |
14 | 9, 13 | mpbiri 167 | . . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 − 1) / 2) ∈ ℕ0) |
15 | 14 | ex 114 | . . 3 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
16 | 2z 9050 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
17 | 16 | a1i 9 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 2 ∈ ℤ) |
18 | nn0z 9042 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
19 | 18 | ad2antrl 481 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ ℤ) |
20 | 2re 8758 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
21 | nn0re 8954 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
22 | ltle 7819 | . . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁)) | |
23 | 20, 21, 22 | sylancr 410 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (2 < 𝑁 → 2 ≤ 𝑁)) |
24 | 23 | adantr 274 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (2 < 𝑁 → 2 ≤ 𝑁)) |
25 | 24 | impcom 124 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 2 ≤ 𝑁) |
26 | eluz2 9300 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
27 | 17, 19, 25, 26 | syl3anbrc 1150 | . . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ (ℤ≥‘2)) |
28 | simprr 506 | . . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 + 1) / 2) ∈ ℕ0) | |
29 | 27, 28 | jca 304 | . . . . 5 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → (𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) |
30 | nno 11530 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ) | |
31 | nnnn0 8952 | . . . . 5 ⊢ (((𝑁 − 1) / 2) ∈ ℕ → ((𝑁 − 1) / 2) ∈ ℕ0) | |
32 | 29, 30, 31 | 3syl 17 | . . . 4 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 − 1) / 2) ∈ ℕ0) |
33 | 32 | ex 114 | . . 3 ⊢ (2 < 𝑁 → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
34 | 15, 33 | jaoi 690 | . 2 ⊢ ((𝑁 = 1 ∨ 2 < 𝑁) → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
35 | 1, 34 | mpcom 36 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 682 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 < clt 7768 ≤ cle 7769 − cmin 7901 / cdiv 8400 ℕcn 8688 2c2 8739 ℕ0cn0 8945 ℤcz 9022 ℤ≥cuz 9294 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-n0 8946 df-z 9023 df-uz 9295 |
This theorem is referenced by: nn0ob 11532 |
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