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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 9003 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℕ0 ↔ ((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0)) | |
2 | nncn 8752 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
3 | 2cnd 8817 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
4 | 2ap0 8837 | . . . . . . . . 9 ⊢ 2 # 0 | |
5 | 4 | a1i 9 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 # 0) |
6 | 2, 3, 5 | diveqap0ad 8584 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 ↔ 𝑁 = 0)) |
7 | eleq1 2203 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ ↔ 0 ∈ ℕ)) | |
8 | 0nnn 8771 | . . . . . . . . . 10 ⊢ ¬ 0 ∈ ℕ | |
9 | 8 | pm2.21i 636 | . . . . . . . . 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 786 | . . . 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 9008 | . 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 698 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 0cc0 7644 # cap 8367 / cdiv 8456 ℕcn 8744 2c2 8795 ℕ0cn0 9001 |
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-3an 965 df-tru 1335 df-fal 1338 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 |
This theorem is referenced by: nnehalf 11637 |
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