ppiublem2 - Metamath Proof Explorer

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Theorem ppiublem2 27181

Description: A prime greater than 3 does not divide 2 or 3, so its residue mod 6 is 1 or 5. (Contributed by Mario Carneiro, 12-Mar-2014.)

**Assertion**
Ref	Expression
ppiublem2	⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ {1, 5})

**Proof of Theorem ppiublem2**
Step	Hyp	Ref	Expression
1		prmz 16649	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
2	1	adantr 479	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
3		6nn 12334	. . . 4 ⊢ 6 ∈ ℕ
4		zmodfz 13894	. . . 4 ⊢ ((𝑃 ∈ ℤ ∧ 6 ∈ ℕ) → (𝑃 mod 6) ∈ (0...(6 − 1)))
5	2, 3, 4	sylancl 584	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ (0...(6 − 1)))
6		6m1e5 12376	. . . 4 ⊢ (6 − 1) = 5
7	6	oveq2i 7430	. . 3 ⊢ (0...(6 − 1)) = (0...5)
8	5, 7	eleqtrdi 2835	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ (0...5))
9		6re 12335	. . . . . . . . . . 11 ⊢ 6 ∈ ℝ
10	9	leidi 11780	. . . . . . . . . 10 ⊢ 6 ≤ 6
11		noel 4330	. . . . . . . . . . . . 13 ⊢ ¬ (𝑃 mod 6) ∈ ∅
12	11	pm2.21i 119	. . . . . . . . . . . 12 ⊢ ((𝑃 mod 6) ∈ ∅ → (𝑃 mod 6) ∈ {1, 5})
13		5lt6 12426	. . . . . . . . . . . . 13 ⊢ 5 < 6
14	3	nnzi 12619	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℤ
15		5nn 12331	. . . . . . . . . . . . . . 15 ⊢ 5 ∈ ℕ
16	15	nnzi 12619	. . . . . . . . . . . . . 14 ⊢ 5 ∈ ℤ
17		fzn 13552	. . . . . . . . . . . . . 14 ⊢ ((6 ∈ ℤ ∧ 5 ∈ ℤ) → (5 < 6 ↔ (6...5) = ∅))
18	14, 16, 17	mp2an 690	. . . . . . . . . . . . 13 ⊢ (5 < 6 ↔ (6...5) = ∅)
19	13, 18	mpbi 229	. . . . . . . . . . . 12 ⊢ (6...5) = ∅
20	12, 19	eleq2s 2843	. . . . . . . . . . 11 ⊢ ((𝑃 mod 6) ∈ (6...5) → (𝑃 mod 6) ∈ {1, 5})
21	20	a1i 11	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (6...5) → (𝑃 mod 6) ∈ {1, 5}))
22	10, 21	pm3.2i 469	. . . . . . . . 9 ⊢ (6 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (6...5) → (𝑃 mod 6) ∈ {1, 5})))
23		5nn0 12525	. . . . . . . . 9 ⊢ 5 ∈ ℕ₀
24		df-6 12312	. . . . . . . . 9 ⊢ 6 = (5 + 1)
25	15	elexi 3482	. . . . . . . . . . 11 ⊢ 5 ∈ V
26	25	prid2 4769	. . . . . . . . . 10 ⊢ 5 ∈ {1, 5}
27	26	3mix3i 1332	. . . . . . . . 9 ⊢ (2 ∥ 5 ∨ 3 ∥ 5 ∨ 5 ∈ {1, 5})
28	22, 23, 24, 27	ppiublem1 27180	. . . . . . . 8 ⊢ (5 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (5...5) → (𝑃 mod 6) ∈ {1, 5})))
29		4nn0 12524	. . . . . . . 8 ⊢ 4 ∈ ℕ₀
30		df-5 12311	. . . . . . . 8 ⊢ 5 = (4 + 1)
31		z4even 16352	. . . . . . . . 9 ⊢ 2 ∥ 4
32	31	3mix1i 1330	. . . . . . . 8 ⊢ (2 ∥ 4 ∨ 3 ∥ 4 ∨ 4 ∈ {1, 5})
33	28, 29, 30, 32	ppiublem1 27180	. . . . . . 7 ⊢ (4 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (4...5) → (𝑃 mod 6) ∈ {1, 5})))
34		3nn0 12523	. . . . . . 7 ⊢ 3 ∈ ℕ₀
35		df-4 12310	. . . . . . 7 ⊢ 4 = (3 + 1)
36		3z 12628	. . . . . . . . 9 ⊢ 3 ∈ ℤ
37		iddvds 16250	. . . . . . . . 9 ⊢ (3 ∈ ℤ → 3 ∥ 3)
38	36, 37	ax-mp 5	. . . . . . . 8 ⊢ 3 ∥ 3
39	38	3mix2i 1331	. . . . . . 7 ⊢ (2 ∥ 3 ∨ 3 ∥ 3 ∨ 3 ∈ {1, 5})
40	33, 34, 35, 39	ppiublem1 27180	. . . . . 6 ⊢ (3 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (3...5) → (𝑃 mod 6) ∈ {1, 5})))
41		2nn0 12522	. . . . . 6 ⊢ 2 ∈ ℕ₀
42		df-3 12309	. . . . . 6 ⊢ 3 = (2 + 1)
43		z2even 16350	. . . . . . 7 ⊢ 2 ∥ 2
44	43	3mix1i 1330	. . . . . 6 ⊢ (2 ∥ 2 ∨ 3 ∥ 2 ∨ 2 ∈ {1, 5})
45	40, 41, 42, 44	ppiublem1 27180	. . . . 5 ⊢ (2 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (2...5) → (𝑃 mod 6) ∈ {1, 5})))
46		1nn0 12521	. . . . 5 ⊢ 1 ∈ ℕ₀
47		df-2 12308	. . . . 5 ⊢ 2 = (1 + 1)
48		1ex 11242	. . . . . . 7 ⊢ 1 ∈ V
49	48	prid1 4768	. . . . . 6 ⊢ 1 ∈ {1, 5}
50	49	3mix3i 1332	. . . . 5 ⊢ (2 ∥ 1 ∨ 3 ∥ 1 ∨ 1 ∈ {1, 5})
51	45, 46, 47, 50	ppiublem1 27180	. . . 4 ⊢ (1 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (1...5) → (𝑃 mod 6) ∈ {1, 5})))
52		0nn0 12520	. . . 4 ⊢ 0 ∈ ℕ₀
53		1e0p1 12752	. . . 4 ⊢ 1 = (0 + 1)
54		z0even 16347	. . . . 5 ⊢ 2 ∥ 0
55	54	3mix1i 1330	. . . 4 ⊢ (2 ∥ 0 ∨ 3 ∥ 0 ∨ 0 ∈ {1, 5})
56	51, 52, 53, 55	ppiublem1 27180	. . 3 ⊢ (0 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (0...5) → (𝑃 mod 6) ∈ {1, 5})))
57	56	simpri 484	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (0...5) → (𝑃 mod 6) ∈ {1, 5}))
58	8, 57	mpd 15	1 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (𝑃 mod 6) ∈ {1, 5})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4322 {cpr 4632 class class class wbr 5149 (class class class)co 7419 0cc0 11140 1c1 11141 < clt 11280 ≤ cle 11281 − cmin 11476 ℕcn 12245 2c2 12300 3c3 12301 4c4 12302 5c5 12303 6c6 12304 ℤcz 12591 ...cfz 13519 mod cmo 13870 ∥ 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-1st 7994 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-inf 9468 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-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fz 13520 df-fl 13793 df-mod 13871 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: ppiub 27182

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