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Mirrors > Home > ILE Home > Th. List > nn0enne | GIF version |
Description: A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.) |
Ref | Expression |
---|---|
nn0enne | ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 ↔ (𝑁 / 2) ∈ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8947 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℕ0 ↔ ((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0)) | |
2 | nncn 8696 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
3 | 2cnd 8761 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
4 | 2ap0 8781 | . . . . . . . . 9 ⊢ 2 # 0 | |
5 | 4 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 # 0) |
6 | 2, 3, 5 | diveqap0ad 8528 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 ↔ 𝑁 = 0)) |
7 | eleq1 2180 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ ↔ 0 ∈ ℕ)) | |
8 | 0nnn 8715 | . . . . . . . . . 10 ⊢ ¬ 0 ∈ ℕ | |
9 | 8 | pm2.21i 620 | . . . . . . . . 9 ⊢ (0 ∈ ℕ → (𝑁 / 2) ∈ ℕ) |
10 | 7, 9 | syl6bi 162 | . . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
11 | 10 | com12 30 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 = 0 → (𝑁 / 2) ∈ ℕ)) |
12 | 6, 11 | sylbid 149 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 → (𝑁 / 2) ∈ ℕ)) |
13 | 12 | com12 30 | . . . . 5 ⊢ ((𝑁 / 2) = 0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
14 | 13 | jao1i 770 | . . . 4 ⊢ (((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0) → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
15 | 1, 14 | sylbi 120 | . . 3 ⊢ ((𝑁 / 2) ∈ ℕ0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
16 | 15 | com12 30 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 → (𝑁 / 2) ∈ ℕ)) |
17 | nnnn0 8952 | . 2 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℕ0) | |
18 | 16, 17 | impbid1 141 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 ↔ (𝑁 / 2) ∈ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∨ wo 682 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 0cc0 7588 # cap 8311 / cdiv 8400 ℕcn 8688 2c2 8739 ℕ0cn0 8945 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 |
This theorem is referenced by: nnehalf 11528 |
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