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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 12245 | . . . 4 ⊢ ((𝑁 / 2) ∈ ℕ0 ↔ ((𝑁 / 2) ∈ ℕ ∨ (𝑁 / 2) = 0)) | |
2 | nncn 11991 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
3 | 2cnd 12061 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
4 | 2ne0 12087 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
5 | 4 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
6 | 2, 3, 5 | diveq0ad 11771 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) = 0 ↔ 𝑁 = 0)) |
7 | eleq1 2826 | . . . . . . . . 9 ⊢ (𝑁 = 0 → (𝑁 ∈ ℕ ↔ 0 ∈ ℕ)) | |
8 | 0nnn 12019 | . . . . . . . . . 10 ⊢ ¬ 0 ∈ ℕ | |
9 | 8 | pm2.21i 119 | . . . . . . . . 9 ⊢ (0 ∈ ℕ → (𝑁 / 2) ∈ ℕ) |
10 | 7, 9 | syl6bi 252 | . . . . . . . 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 855 | . . . 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 12250 | . 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 844 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7267 0cc0 10881 / cdiv 11642 ℕcn 11983 2c2 12038 ℕ0cn0 12243 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-n0 12244 |
This theorem is referenced by: nnehalf 16098 blennn0em1 45915 blennn0e2 45918 |
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