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Mirrors > Home > MPE Home > Th. List > oddprm | Structured version Visualization version GIF version |
Description: A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.) |
Ref | Expression |
---|---|
oddprm | ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4054 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ) | |
2 | prmz 16009 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ) |
4 | eldifsni 4683 | . . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2) | |
5 | 4 | necomd 3042 | . . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁) |
6 | 5 | neneqd 2992 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁) |
7 | 2z 12002 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
8 | uzid 12246 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
10 | dvdsprm 16037 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℙ) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) | |
11 | 9, 1, 10 | sylancr 590 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) |
12 | 6, 11 | mtbird 328 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁) |
13 | 1z 12000 | . . . . 5 ⊢ 1 ∈ ℤ | |
14 | n2dvds1 15709 | . . . . 5 ⊢ ¬ 2 ∥ 1 | |
15 | omoe 15705 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) ∧ (1 ∈ ℤ ∧ ¬ 2 ∥ 1)) → 2 ∥ (𝑁 − 1)) | |
16 | 13, 14, 15 | mpanr12 704 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1)) |
17 | 3, 12, 16 | syl2anc 587 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ∥ (𝑁 − 1)) |
18 | prmnn 16008 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
19 | nnm1nn0 11926 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
20 | 1, 18, 19 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ0) |
21 | nn0z 11993 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) | |
22 | evend2 15698 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
23 | 20, 21, 22 | 3syl 18 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) |
24 | 17, 23 | mpbid 235 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℤ) |
25 | prmuz2 16030 | . . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ≥‘2)) | |
26 | uz2m1nn 12311 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | |
27 | nngt0 11656 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1)) | |
28 | nnre 11632 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ) | |
29 | 2rp 12382 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ+) |
31 | 28, 30 | gt0divd 12456 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) |
32 | 27, 31 | mpbid 235 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < ((𝑁 − 1) / 2)) |
33 | 1, 25, 26, 32 | 4syl 19 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 0 < ((𝑁 − 1) / 2)) |
34 | elnnz 11979 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2))) | |
35 | 24, 33, 34 | sylanbrc 586 | 1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 < clt 10664 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ℝ+crp 12377 ∥ cdvds 15599 ℙcprime 16005 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 |
This theorem is referenced by: nnoddn2prm 16138 4sqlem19 16289 lgslem1 25881 lgslem4 25884 lgsval2lem 25891 lgsvalmod 25900 lgsmod 25907 lgsdirprm 25915 lgsne0 25919 lgsqrlem1 25930 lgsqrlem2 25931 lgsqrlem3 25932 lgsqrlem4 25933 gausslemma2dlem4 25953 lgseisenlem1 25959 lgseisenlem2 25960 lgseisenlem4 25962 lgseisen 25963 m1lgs 25972 2lgslem2 25979 fmtnoprmfac2 44084 |
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