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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 4123 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ) | |
2 | prmz 16649 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ) |
4 | eldifsni 4795 | . . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2) | |
5 | 4 | necomd 2985 | . . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁) |
6 | 5 | neneqd 2934 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁) |
7 | 2z 12627 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
8 | uzid 12870 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
10 | dvdsprm 16677 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℙ) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) | |
11 | 9, 1, 10 | sylancr 585 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) |
12 | 6, 11 | mtbird 324 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁) |
13 | 1z 12625 | . . . . 5 ⊢ 1 ∈ ℤ | |
14 | n2dvds1 16348 | . . . . 5 ⊢ ¬ 2 ∥ 1 | |
15 | omoe 16344 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) ∧ (1 ∈ ℤ ∧ ¬ 2 ∥ 1)) → 2 ∥ (𝑁 − 1)) | |
16 | 13, 14, 15 | mpanr12 703 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1)) |
17 | 3, 12, 16 | syl2anc 582 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ∥ (𝑁 − 1)) |
18 | prmnn 16648 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
19 | nnm1nn0 12546 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
20 | 1, 18, 19 | 3syl 18 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ0) |
21 | nn0z 12616 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) | |
22 | evend2 16337 | . . . 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 231 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℤ) |
25 | prmuz2 16670 | . . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ≥‘2)) | |
26 | uz2m1nn 12940 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | |
27 | nngt0 12276 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1)) | |
28 | nnre 12252 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ) | |
29 | 2rp 13014 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ+) |
31 | 28, 30 | gt0divd 13088 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) |
32 | 27, 31 | mpbid 231 | . . 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 12601 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2))) | |
35 | 24, 33, 34 | sylanbrc 581 | 1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 {csn 4630 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 0cc0 11140 1c1 11141 < clt 11280 − cmin 11476 / cdiv 11903 ℕcn 12245 2c2 12300 ℕ0cn0 12505 ℤcz 12591 ℤ≥cuz 12855 ℝ+crp 13009 ∥ cdvds 16234 ℙcprime 16645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-prm 16646 |
This theorem is referenced by: nnoddn2prm 16783 4sqlem19 16935 lgslem1 27275 lgslem4 27278 lgsval2lem 27285 lgsvalmod 27294 lgsmod 27301 lgsdirprm 27309 lgsne0 27313 lgsqrlem1 27324 lgsqrlem2 27325 lgsqrlem3 27326 lgsqrlem4 27327 gausslemma2dlem4 27347 lgseisenlem1 27353 lgseisenlem2 27354 lgseisenlem4 27356 lgseisen 27357 m1lgs 27366 2lgslem2 27373 fmtnoprmfac2 47044 |
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