ppiublem1 - Metamath Proof Explorer

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Theorem ppiublem1 27328

Description: Lemma for ppiub 27330. (Contributed by Mario Carneiro, 12-Mar-2014.)

**Hypotheses**
Ref	Expression
ppiublem1.1	⊢ (𝑁 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5})))
ppiublem1.2	⊢ 𝑀 ∈ ℕ₀
ppiublem1.3	⊢ 𝑁 = (𝑀 + 1)
ppiublem1.4	⊢ (2 ∥ 𝑀 ∨ 3 ∥ 𝑀 ∨ 𝑀 ∈ {1, 5})

**Assertion**
Ref	Expression
ppiublem1	⊢ (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

**Proof of Theorem ppiublem1**
Step	Hyp	Ref	Expression
1		ppiublem1.1	. . . . . 6 ⊢ (𝑁 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5})))
2	1	simpli 488	. . . . 5 ⊢ 𝑁 ≤ 6
3		ppiublem1.3	. . . . 5 ⊢ 𝑁 = (𝑀 + 1)
4		df-6 12303	. . . . 5 ⊢ 6 = (5 + 1)
5	2, 3, 4	3brtr3i 5141	. . . 4 ⊢ (𝑀 + 1) ≤ (5 + 1)
6		ppiublem1.2	. . . . . 6 ⊢ 𝑀 ∈ ℕ₀
7	6	nn0rei 12511	. . . . 5 ⊢ 𝑀 ∈ ℝ
8		5re 12324	. . . . 5 ⊢ 5 ∈ ℝ
9		1re 11204	. . . . 5 ⊢ 1 ∈ ℝ
10	7, 8, 9	leadd1i 11765	. . . 4 ⊢ (𝑀 ≤ 5 ↔ (𝑀 + 1) ≤ (5 + 1))
11	5, 10	mpbir 234	. . 3 ⊢ 𝑀 ≤ 5
12		6re 12327	. . . 4 ⊢ 6 ∈ ℝ
13		5lt6 12420	. . . 4 ⊢ 5 < 6
14	8, 12, 13	ltleii 11329	. . 3 ⊢ 5 ≤ 6
15	7, 8, 12	letri 11335	. . 3 ⊢ ((𝑀 ≤ 5 ∧ 5 ≤ 6) → 𝑀 ≤ 6)
16	11, 14, 15	mp2an 704	. 2 ⊢ 𝑀 ≤ 6
17	6	nn0zi 12615	. . . . 5 ⊢ 𝑀 ∈ ℤ
18		5nn 12323	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12614	. . . . 5 ⊢ 5 ∈ ℤ
20		eluz2 12864	. . . . 5 ⊢ (5 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 5 ∈ ℤ ∧ 𝑀 ≤ 5))
21	17, 19, 11, 20	mpbir3an 1358	. . . 4 ⊢ 5 ∈ (ℤ_≥‘𝑀)
22		elfzp12 13627	. . . 4 ⊢ (5 ∈ (ℤ_≥‘𝑀) → ((𝑃 mod 6) ∈ (𝑀...5) ↔ ((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))))
23	21, 22	ax-mp 5	. . 3 ⊢ ((𝑃 mod 6) ∈ (𝑀...5) ↔ ((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5)))
24		ppiublem1.4	. . . . 5 ⊢ (2 ∥ 𝑀 ∨ 3 ∥ 𝑀 ∨ 𝑀 ∈ {1, 5})
25		2nn 12310	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
26		6nn 12326	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
27		prmz 16729	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
28	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
29		3z 12623	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
30		2z 12622	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
31		dvdsmul2 16332	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (3 · 2))
32	29, 30, 31	mp2an 704	. . . . . . . . . . . . 13 ⊢ 2 ∥ (3 · 2)
33		3t2e6 12402	. . . . . . . . . . . . 13 ⊢ (3 · 2) = 6
34	32, 33	breqtri 5137	. . . . . . . . . . . 12 ⊢ 2 ∥ 6
35		dvdsmod 16383	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 2 ∥ 6) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
36	34, 35	mpan2 703	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
37	25, 26, 28, 36	mp3an12i 1491	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
38		uzid 12873	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
39	30, 38	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
40		simpl 487	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℙ)
41		dvdsprm 16758	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
42	39, 40, 41	sylancr 598	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
43	37, 42	bitrd 282	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 = 𝑃))
44		simpr 489	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 4 ≤ 𝑃)
45		breq2 5114	. . . . . . . . . . 11 ⊢ (2 = 𝑃 → (4 ≤ 2 ↔ 4 ≤ 𝑃))
46	44, 45	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → 4 ≤ 2))
47		2lt4 12414	. . . . . . . . . . . 12 ⊢ 2 < 4
48		2re 12311	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
49		4re 12321	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
50	48, 49	ltnlei 11327	. . . . . . . . . . . 12 ⊢ (2 < 4 ↔ ¬ 4 ≤ 2)
51	47, 50	mpbi 233	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 2
52	51	pm2.21i 120	. . . . . . . . . 10 ⊢ (4 ≤ 2 → (𝑃 mod 6) ∈ {1, 5})
53	46, 52	syl6 36	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
54	43, 53	sylbid 243	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
55		breq2 5114	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑀))
56	55	imbi1d 344	. . . . . . . 8 ⊢ ((𝑃 mod 6) = 𝑀 → ((2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}) ↔ (2 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
57	54, 56	syl5ibcom 248	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (2 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
58	57	com3r 88	. . . . . 6 ⊢ (2 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
59		3nn 12316	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
60		dvdsmul1 16331	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
61	29, 30, 60	mp2an 704	. . . . . . . . . . . . 13 ⊢ 3 ∥ (3 · 2)
62	61, 33	breqtri 5137	. . . . . . . . . . . 12 ⊢ 3 ∥ 6
63		dvdsmod 16383	. . . . . . . . . . . 12 ⊢ (((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 3 ∥ 6) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
64	62, 63	mpan2 703	. . . . . . . . . . 11 ⊢ ((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
65	59, 26, 28, 64	mp3an12i 1491	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
66		df-3 12300	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
67		peano2uz 12921	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘2) → (2 + 1) ∈ (ℤ_≥‘2))
68	39, 67	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2 + 1) ∈ (ℤ_≥‘2)
69	66, 68	eqeltri 2865	. . . . . . . . . . 11 ⊢ 3 ∈ (ℤ_≥‘2)
70		dvdsprm 16758	. . . . . . . . . . 11 ⊢ ((3 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
71	69, 40, 70	sylancr 598	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
72	65, 71	bitrd 282	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 = 𝑃))
73		breq2 5114	. . . . . . . . . . 11 ⊢ (3 = 𝑃 → (4 ≤ 3 ↔ 4 ≤ 𝑃))
74	44, 73	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → 4 ≤ 3))
75		3lt4 12413	. . . . . . . . . . . 12 ⊢ 3 < 4
76		3re 12317	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
77	76, 49	ltnlei 11327	. . . . . . . . . . . 12 ⊢ (3 < 4 ↔ ¬ 4 ≤ 3)
78	75, 77	mpbi 233	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 3
79	78	pm2.21i 120	. . . . . . . . . 10 ⊢ (4 ≤ 3 → (𝑃 mod 6) ∈ {1, 5})
80	74, 79	syl6 36	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
81	72, 80	sylbid 243	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
82		breq2 5114	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑀))
83	82	imbi1d 344	. . . . . . . 8 ⊢ ((𝑃 mod 6) = 𝑀 → ((3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}) ↔ (3 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
84	81, 83	syl5ibcom 248	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (3 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
85	84	com3r 88	. . . . . 6 ⊢ (3 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
86		eleq1a 2864	. . . . . . 7 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5}))
87	86	a1d 26	. . . . . 6 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
88	58, 85, 87	3jaoi 1452	. . . . 5 ⊢ ((2 ∥ 𝑀 ∨ 3 ∥ 𝑀 ∨ 𝑀 ∈ {1, 5}) → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
89	24, 88	ax-mp 5	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5}))
90	3	oveq1i 7418	. . . . . 6 ⊢ (𝑁...5) = ((𝑀 + 1)...5)
91	90	eleq2i 2861	. . . . 5 ⊢ ((𝑃 mod 6) ∈ (𝑁...5) ↔ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))
92	1	simpri 490	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5}))
93	91, 92	biimtrrid 246	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ ((𝑀 + 1)...5) → (𝑃 mod 6) ∈ {1, 5}))
94	89, 93	jaod 872	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5)) → (𝑃 mod 6) ∈ {1, 5}))
95	23, 94	biimtrid 245	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5}))
96	16, 95	pm3.2i 475	1 ⊢ (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 {cpr 4593 class class class wbr 5110 ‘cfv 6533 (class class class)co 7408 1c1 11097 + caddc 11099 · cmul 11101 < clt 11239 ≤ cle 11240 ℕcn 12229 2c2 12291 3c3 12292 4c4 12293 5c5 12294 6c6 12295 ℕ₀cn0 12500 ℤcz 12587 ℤ_≥cuz 12858 ...cfz 13531 mod cmo 13898 ∥ cdvds 16306 ℙcprime 16725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12501 df-z 12588 df-uz 12859 df-rp 13013 df-fz 13532 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-dvds 16307 df-prm 16726

This theorem is referenced by: ppiublem2 27329

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