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Mirrors > Home > ILE Home > Th. List > oddprm | 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 3230 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ) | |
2 | prmz 12004 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ) |
4 | eldifsni 3690 | . . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2) | |
5 | 4 | necomd 2413 | . . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁) |
6 | 5 | neneqd 2348 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁) |
7 | 2z 9201 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
8 | uzid 9459 | . . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
9 | 7, 8 | ax-mp 5 | . . . . . 6 ⊢ 2 ∈ (ℤ≥‘2) |
10 | dvdsprm 12030 | . . . . . 6 ⊢ ((2 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℙ) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) | |
11 | 9, 1, 10 | sylancr 411 | . . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ 𝑁 ↔ 2 = 𝑁)) |
12 | 6, 11 | mtbird 663 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁) |
13 | 1z 9199 | . . . . 5 ⊢ 1 ∈ ℤ | |
14 | n2dvds1 11816 | . . . . 5 ⊢ ¬ 2 ∥ 1 | |
15 | omoe 11800 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) ∧ (1 ∈ ℤ ∧ ¬ 2 ∥ 1)) → 2 ∥ (𝑁 − 1)) | |
16 | 13, 14, 15 | mpanr12 436 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1)) |
17 | 3, 12, 16 | syl2anc 409 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ∥ (𝑁 − 1)) |
18 | prmnn 12003 | . . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ) | |
19 | nnm1nn0 9137 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
20 | 1, 18, 19 | 3syl 17 | . . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ0) |
21 | nn0z 9193 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ0 → (𝑁 − 1) ∈ ℤ) | |
22 | evend2 11793 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) | |
23 | 20, 21, 22 | 3syl 17 | . . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 2) ∈ ℤ)) |
24 | 17, 23 | mpbid 146 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℤ) |
25 | prmuz2 12024 | . . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ≥‘2)) | |
26 | uz2m1nn 9522 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | |
27 | nngt0 8864 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1)) | |
28 | nnre 8846 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ) | |
29 | 2rp 9572 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
30 | 29 | a1i 9 | . . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ+) |
31 | 28, 30 | gt0divd 9648 | . . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2))) |
32 | 27, 31 | mpbid 146 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < ((𝑁 − 1) / 2)) |
33 | 1, 25, 26, 32 | 4syl 18 | . 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 0 < ((𝑁 − 1) / 2)) |
34 | elnnz 9183 | . 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2))) | |
35 | 24, 33, 34 | sylanbrc 414 | 1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∖ cdif 3099 {csn 3561 class class class wbr 3967 ‘cfv 5173 (class class class)co 5827 0cc0 7735 1c1 7736 < clt 7915 − cmin 8051 / cdiv 8550 ℕcn 8839 2c2 8890 ℕ0cn0 9096 ℤcz 9173 ℤ≥cuz 9445 ℝ+crp 9567 ∥ cdvds 11695 ℙcprime 12000 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-xor 1358 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-1o 6366 df-2o 6367 df-er 6483 df-en 6689 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-seqfrec 10355 df-exp 10429 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-dvds 11696 df-prm 12001 |
This theorem is referenced by: nnoddn2prm 12151 |
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