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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 12005 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
2 | oddp1d2 15707 | . . 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 11650 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
5 | 4 | nnred 11653 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ) |
6 | 2re 11712 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
8 | nnre 11645 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
9 | 1red 10642 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
10 | nngt0 11669 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁) | |
11 | 0lt1 11162 | . . . . . . . . 9 ⊢ 0 < 1 | |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 0 < 1) |
13 | 8, 9, 10, 12 | addgt0d 11215 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1)) |
14 | 2pos 11741 | . . . . . . . 8 ⊢ 0 < 2 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
16 | 5, 7, 13, 15 | divgt0d 11575 | . . . . . 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 11992 | . . . . 5 ⊢ (((𝑁 + 1) / 2) ∈ ℕ ↔ (((𝑁 + 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 + 1) / 2))) | |
19 | 17, 18 | sylibr 236 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → ((𝑁 + 1) / 2) ∈ ℕ) |
20 | 19 | ex 415 | . . 3 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℕ)) |
21 | nnz 12005 | . . 3 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ) | |
22 | 20, 21 | impbid1 227 | . 2 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
23 | 3, 22 | bitrd 281 | 1 ⊢ (𝑁 ∈ ℕ → (¬ 2 ∥ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 / cdiv 11297 ℕcn 11638 2c2 11693 ℤcz 11982 ∥ cdvds 15607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-dvds 15608 |
This theorem is referenced by: gausslemma2dlem0b 25933 |
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