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Mirrors > Home > MPE Home > Th. List > nn0enne | Structured version Visualization version 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 12349 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℕ0 ↔ ((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0)) | |
2 | nncn 12095 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
3 | 2cnd 12165 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
4 | 2ne0 12191 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
6 | 2, 3, 5 | diveq0ad 11875 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 ↔ 𝑁 = 0)) |
7 | eleq1 2826 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ ↔ 0 ∈ ℕ)) | |
8 | 0nnn 12123 | . . . . . . . . . 10 ⊢ ¬ 0 ∈ ℕ | |
9 | 8 | pm2.21i 119 | . . . . . . . . 9 ⊢ (0 ∈ ℕ → (𝑁 / 2) ∈ ℕ) |
10 | 7, 9 | syl6bi 253 | . . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
11 | 10 | com12 32 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (𝑁 = 0 → (𝑁 / 2) ∈ ℕ)) |
12 | 6, 11 | sylbid 239 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 → (𝑁 / 2) ∈ ℕ)) |
13 | 12 | com12 32 | . . . . 5 ⊢ ((𝑁 / 2) = 0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
14 | 13 | jao1i 857 | . . . 4 ⊢ (((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0) → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
15 | 1, 14 | sylbi 216 | . . 3 ⊢ ((𝑁 / 2) ∈ ℕ0 → (𝑁 ∈ ℕ → (𝑁 / 2) ∈ ℕ)) |
16 | 15 | com12 32 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 → (𝑁 / 2) ∈ ℕ)) |
17 | nnnn0 12354 | . 2 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℕ0) | |
18 | 16, 17 | impbid1 224 | 1 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ0 ↔ (𝑁 / 2) ∈ ℕ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 (class class class)co 7350 0cc0 10985 / cdiv 11746 ℕcn 12087 2c2 12142 ℕ0cn0 12347 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-n0 12348 |
This theorem is referenced by: nnehalf 16196 blennn0em1 46377 blennn0e2 46380 |
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