oddprm - Intuitionistic Logic Explorer

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Theorem oddprm 12797

Description: A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 10-Jul-2022.)

**Assertion**
Ref	Expression
oddprm	⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ)

**Proof of Theorem oddprm**
Step	Hyp	Ref	Expression
1		eldifi 3326	. . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ)
2		prmz 12648	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ)
3	1, 2	syl 14	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ)
4		eldifsni 3797	. . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2)
5	4	necomd 2486	. . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁)
6	5	neneqd 2421	. . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁)
7		2z 9485	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 9748	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		dvdsprm 12674	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℙ) → (2 ∥ 𝑁 ↔ 2 = 𝑁))
11	9, 1, 10	sylancr 414	. . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (2 ∥ 𝑁 ↔ 2 = 𝑁))
12	6, 11	mtbird 677	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁)
13		1z 9483	. . . . 5 ⊢ 1 ∈ ℤ
14		n2dvds1 12438	. . . . 5 ⊢ ¬ 2 ∥ 1
15		omoe 12422	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) ∧ (1 ∈ ℤ ∧ ¬ 2 ∥ 1)) → 2 ∥ (𝑁 − 1))
16	13, 14, 15	mpanr12 439	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ ¬ 2 ∥ 𝑁) → 2 ∥ (𝑁 − 1))
17	3, 12, 16	syl2anc 411	. . 3 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ∥ (𝑁 − 1))
18		prmnn 12647	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
19		nnm1nn0 9421	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
20	1, 18, 19	3syl 17	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ₀)
21		nn0z 9477	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (𝑁 − 1) ∈ ℤ)
22		evend2 12415	. . . 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 147	. 2 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℤ)
25		prmuz2 12668	. . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
26		uz2m1nn 9812	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
27		nngt0 9146	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1))
28		nnre 9128	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ)
29		2rp 9866	. . . . . 6 ⊢ 2 ∈ ℝ⁺
30	29	a1i 9	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ⁺)
31	28, 30	gt0divd 9942	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → (0 < (𝑁 − 1) ↔ 0 < ((𝑁 − 1) / 2)))
32	27, 31	mpbid 147	. . 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 9467	. 2 ⊢ (((𝑁 − 1) / 2) ∈ ℕ ↔ (((𝑁 − 1) / 2) ∈ ℤ ∧ 0 < ((𝑁 − 1) / 2)))
35	24, 33, 34	sylanbrc 417	1 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ((𝑁 − 1) / 2) ∈ ℕ)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1395 ∈ wcel 2200 ∖ cdif 3194 {csn 3666 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 0cc0 8010 1c1 8011 < clt 8192 − cmin 8328 / cdiv 8830 ℕcn 9121 2c2 9172 ℕ₀cn0 9380 ℤcz 9457 ℤ_≥cuz 9733 ℝ⁺crp 9861 ∥ cdvds 12313 ℙcprime 12644

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-2o 6569 df-er 6688 df-en 6896 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-dvds 12314 df-prm 12645

This theorem is referenced by: nnoddn2prm 12798 4sqlem19 12947 lgslem1 15694 lgslem4 15697 lgsval2lem 15704 lgsvalmod 15713 lgsmod 15720 lgsdirprm 15728 lgsne0 15732 gausslemma2dlem4 15758 lgseisenlem1 15764 lgseisenlem2 15765 lgseisenlem4 15767 lgseisen 15768 m1lgs 15779 2lgslem1 15785 2lgslem2 15786

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