![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9097 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | oddp1d2 11623 | . . 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 8756 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | nnred 8757 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ) |
6 | 2re 8814 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
8 | nnre 8751 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 7805 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
10 | nngt0 8769 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
11 | 0lt1 7913 | . . . . . . . . . 10 ⊢ 0 < 1 | |
12 | 11 | a1i 9 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 0 < 1) |
13 | 8, 9, 10, 12 | addgt0d 8307 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1)) |
14 | 2pos 8835 | . . . . . . . . 9 ⊢ 0 < 2 | |
15 | 14 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
16 | 5, 7, 13, 15 | divgt0d 8717 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < ((𝑁 + 1) / 2)) |
17 | 16 | anim1i 338 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (0 < ((𝑁 + 1) / 2) ∧ ((𝑁 + 1) / 2) ∈ ℤ)) |
18 | 17 | ancomd 265 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) |
19 | elnnz 9088 | . . . . 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 9097 | . . 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 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 0cc0 7644 1c1 7645 + caddc 7647 < clt 7824 / cdiv 8456 ℕcn 8744 2c2 8795 ℤcz 9078 ∥ cdvds 11529 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-xor 1355 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-n0 9002 df-z 9079 df-dvds 11530 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |