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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 11827 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (𝑁 = 1 ∨ 2 < 𝑁)) | |
2 | 1m1e0 8917 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
3 | 2 | oveq1i 5846 | . . . . . . 7 ⊢ ((1 − 1) / 2) = (0 / 2) |
4 | 2cn 8919 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
5 | 2ap0 8941 | . . . . . . . 8 ⊢ 2 # 0 | |
6 | 4, 5 | div0api 8633 | . . . . . . 7 ⊢ (0 / 2) = 0 |
7 | 3, 6 | eqtri 2185 | . . . . . 6 ⊢ ((1 − 1) / 2) = 0 |
8 | 0nn0 9120 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
9 | 7, 8 | eqeltri 2237 | . . . . 5 ⊢ ((1 − 1) / 2) ∈ ℕ0 |
10 | oveq1 5843 | . . . . . . . 8 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
11 | 10 | oveq1d 5851 | . . . . . . 7 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = ((1 − 1) / 2)) |
12 | 11 | eleq1d 2233 | . . . . . 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 9210 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
17 | 16 | a1i 9 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 2 ∈ ℤ) |
18 | nn0z 9202 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
19 | 18 | ad2antrl 482 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ ℤ) |
20 | 2re 8918 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
21 | nn0re 9114 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
22 | ltle 7977 | . . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁)) | |
23 | 20, 21, 22 | sylancr 411 | . . . . . . . . 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 9463 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
27 | 17, 19, 25, 26 | syl3anbrc 1170 | . . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ (ℤ≥‘2)) |
28 | simprr 522 | . . . . . 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 11828 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ) | |
31 | nnnn0 9112 | . . . . 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 706 | . 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 698 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 1c1 7745 + caddc 7747 < clt 7924 ≤ cle 7925 − cmin 8060 / cdiv 8559 ℕcn 8848 2c2 8899 ℕ0cn0 9105 ℤcz 9182 ℤ≥cuz 9457 |
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-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 |
This theorem is referenced by: nn0ob 11830 |
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