oddprm - Intuitionistic Logic Explorer

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Theorem oddprm 12957

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 3341	. . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℙ)
2		prmz 12808	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ)
3	1, 2	syl 14	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ ℤ)
4		eldifsni 3822	. . . . . . 7 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ≠ 2)
5	4	necomd 2498	. . . . . 6 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑁)
6	5	neneqd 2433	. . . . 5 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 = 𝑁)
7		2z 9605	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 9868	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		dvdsprm 12834	. . . . . 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 680	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑁)
13		1z 9603	. . . . 5 ⊢ 1 ∈ ℤ
14		n2dvds1 12598	. . . . 5 ⊢ ¬ 2 ∥ 1
15		omoe 12582	. . . . 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 12807	. . . . 5 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
19		nnm1nn0 9537	. . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
20	1, 18, 19	3syl 17	. . . 4 ⊢ (𝑁 ∈ (ℙ ∖ {2}) → (𝑁 − 1) ∈ ℕ₀)
21		nn0z 9597	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ₀ → (𝑁 − 1) ∈ ℤ)
22		evend2 12575	. . . 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 12828	. . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ (ℤ_≥‘2))
26		uz2m1nn 9937	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
27		nngt0 9262	. . . 4 ⊢ ((𝑁 − 1) ∈ ℕ → 0 < (𝑁 − 1))
28		nnre 9244	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → (𝑁 − 1) ∈ ℝ)
29		2rp 9991	. . . . . 6 ⊢ 2 ∈ ℝ⁺
30	29	a1i 9	. . . . 5 ⊢ ((𝑁 − 1) ∈ ℕ → 2 ∈ ℝ⁺)
31	28, 30	gt0divd 10067	. . . 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 9587	. 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 1398 ∈ wcel 2203 ∖ cdif 3208 {csn 3689 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 0cc0 8127 1c1 8128 < clt 8308 − cmin 8444 / cdiv 8946 ℕcn 9237 2c2 9288 ℕ₀cn0 9496 ℤcz 9577 ℤ_≥cuz 9853 ℝ⁺crp 9986 ∥ cdvds 12473 ℙcprime 12804

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-1o 6647 df-2o 6648 df-er 6767 df-en 6976 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-dvds 12474 df-prm 12805

This theorem is referenced by: nnoddn2prm 12958 4sqlem19 13107 lgslem1 15873 lgslem4 15876 lgsval2lem 15883 lgsvalmod 15892 lgsmod 15899 lgsdirprm 15907 lgsne0 15911 gausslemma2dlem4 15937 lgseisenlem1 15943 lgseisenlem2 15944 lgseisenlem4 15946 lgseisen 15947 m1lgs 15958 2lgslem1 15964 2lgslem2 15965

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