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Mirrors > Home > ILE Home > Th. List > nnoddm1d2 | GIF version |
Description: A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.) |
Ref | Expression |
---|---|
nnoddm1d2 | ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 8867 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | oddp1d2 11317 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
4 | peano2nn 8532 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | nnred 8533 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ) |
6 | 2re 8590 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
8 | nnre 8527 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 7600 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
10 | nngt0 8545 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
11 | 0lt1 7707 | . . . . . . . . . 10 ⊢ 0 < 1 | |
12 | 11 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 < 1) |
13 | 8, 9, 10, 12 | addgt0d 8095 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1)) |
14 | 2pos 8611 | . . . . . . . . 9 ⊢ 0 < 2 | |
15 | 14 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
16 | 5, 7, 13, 15 | divgt0d 8493 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < ((𝑁 + 1) / 2)) |
17 | 16 | anim1i 334 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (0 < ((𝑁 + 1) / 2) ∧ ((𝑁 + 1) / 2) ∈ ℤ)) |
18 | 17 | ancomd 264 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) |
19 | elnnz 8858 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) | |
20 | 18, 19 | sylibr 133 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ) |
21 | 20 | ex 114 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ)) |
22 | nnz 8867 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
23 | 21, 22 | impbid1 141 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
24 | 3, 23 | bitrd 187 | 1 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 ℝcr 7446 0cc0 7447 1c1 7448 + caddc 7450 < clt 7619 / cdiv 8236 ℕcn 8520 2c2 8571 ℤcz 8848 ∥ cdvds 11223 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-xor 1319 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-n0 8772 df-z 8849 df-dvds 11224 |
This theorem is referenced by: (None) |
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