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Mirrors > Home > MPE Home > Th. List > nn0o | Structured version Visualization version GIF version |
Description: An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.) |
Ref | Expression |
---|---|
nn0o | ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0o1gt2 16255 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (𝑁 = 1 ∨ 2 < 𝑁)) | |
2 | 1m1e0 12221 | . . . . . . . 8 ⊢ (1 − 1) = 0 | |
3 | 2 | oveq1i 7363 | . . . . . . 7 ⊢ ((1 − 1) / 2) = (0 / 2) |
4 | 2cn 12224 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
5 | 2ne0 12253 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
6 | 4, 5 | div0i 11885 | . . . . . . 7 ⊢ (0 / 2) = 0 |
7 | 3, 6 | eqtri 2764 | . . . . . 6 ⊢ ((1 − 1) / 2) = 0 |
8 | 0nn0 12424 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
9 | 7, 8 | eqeltri 2834 | . . . . 5 ⊢ ((1 − 1) / 2) ∈ ℕ0 |
10 | oveq1 7360 | . . . . . . . 8 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1)) | |
11 | 10 | oveq1d 7368 | . . . . . . 7 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = ((1 − 1) / 2)) |
12 | 11 | eleq1d 2822 | . . . . . 6 ⊢ (𝑁 = 1 → (((𝑁 − 1) / 2) ∈ ℕ0 ↔ ((1 − 1) / 2) ∈ ℕ0)) |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → (((𝑁 − 1) / 2) ∈ ℕ0 ↔ ((1 − 1) / 2) ∈ ℕ0)) |
14 | 9, 13 | mpbiri 257 | . . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 − 1) / 2) ∈ ℕ0) |
15 | 14 | ex 413 | . . 3 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
16 | 2z 12531 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 2 ∈ ℤ) |
18 | nn0z 12520 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
19 | 18 | ad2antrl 726 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ ℤ) |
20 | 2re 12223 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
21 | nn0re 12418 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
22 | ltle 11239 | . . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁)) | |
23 | 20, 21, 22 | sylancr 587 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (2 < 𝑁 → 2 ≤ 𝑁)) |
24 | 23 | adantr 481 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (2 < 𝑁 → 2 ≤ 𝑁)) |
25 | 24 | impcom 408 | . . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 2 ≤ 𝑁) |
26 | eluz2 12765 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁)) | |
27 | 17, 19, 25, 26 | syl3anbrc 1343 | . . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → 𝑁 ∈ (ℤ≥‘2)) |
28 | simprr 771 | . . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 + 1) / 2) ∈ ℕ0) | |
29 | 27, 28 | jca 512 | . . . . 5 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → (𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) |
30 | nno 16256 | . . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ) | |
31 | nnnn0 12416 | . . . . 5 ⊢ (((𝑁 − 1) / 2) ∈ ℕ → ((𝑁 − 1) / 2) ∈ ℕ0) | |
32 | 29, 30, 31 | 3syl 18 | . . . 4 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0)) → ((𝑁 − 1) / 2) ∈ ℕ0) |
33 | 32 | ex 413 | . . 3 ⊢ (2 < 𝑁 → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
34 | 15, 33 | jaoi 855 | . 2 ⊢ ((𝑁 = 1 ∨ 2 < 𝑁) → ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0)) |
35 | 1, 34 | mpcom 38 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 < clt 11185 ≤ cle 11186 − cmin 11381 / cdiv 11808 ℕcn 12149 2c2 12204 ℕ0cn0 12409 ℤcz 12495 ℤ≥cuz 12759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 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 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 |
This theorem is referenced by: nn0ob 16258 nn0onn0ex 46541 nneom 46545 flnn0div2ge 46551 flnn0ohalf 46552 |
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