![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nnoddm1d2 | Structured version Visualization version 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 12527 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | oddp1d2 16247 | . . 3 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℤ)) |
4 | peano2nn 12172 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | nnred 12175 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ) |
6 | 2re 12234 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
8 | nnre 12167 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 11163 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
10 | nngt0 12191 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
11 | 0lt1 11684 | . . . . . . . . 9 ⊢ 0 < 1 | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1) |
13 | 8, 9, 10, 12 | addgt0d 11737 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1)) |
14 | 2pos 12263 | . . . . . . . 8 ⊢ 0 < 2 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
16 | 5, 7, 13, 15 | divgt0d 12097 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 0 < ((𝑁 + 1) / 2)) |
17 | 16 | anim1ci 617 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) |
18 | elnnz 12516 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) | |
19 | 17, 18 | sylibr 233 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ) |
20 | 19 | ex 414 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ)) |
21 | nnz 12527 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
22 | 20, 21 | impbid1 224 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
23 | 3, 22 | bitrd 279 | 1 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5110 (class class class)co 7362 ℝcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 / cdiv 11819 ℕcn 12160 2c2 12215 ℤcz 12506 ∥ cdvds 16143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-dvds 16144 |
This theorem is referenced by: gausslemma2dlem0b 26721 |
Copyright terms: Public domain | W3C validator |