ppiublem1 - Metamath Proof Explorer

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Theorem ppiublem1 27181

Description: Lemma for ppiub 27183. (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 483	. . . . 5 ⊢ 𝑁 ≤ 6
3		ppiublem1.3	. . . . 5 ⊢ 𝑁 = (𝑀 + 1)
4		df-6 12224	. . . . 5 ⊢ 6 = (5 + 1)
5	2, 3, 4	3brtr3i 5129	. . . 4 ⊢ (𝑀 + 1) ≤ (5 + 1)
6		ppiublem1.2	. . . . . 6 ⊢ 𝑀 ∈ ℕ₀
7	6	nn0rei 12424	. . . . 5 ⊢ 𝑀 ∈ ℝ
8		5re 12244	. . . . 5 ⊢ 5 ∈ ℝ
9		1re 11144	. . . . 5 ⊢ 1 ∈ ℝ
10	7, 8, 9	leadd1i 11704	. . . 4 ⊢ (𝑀 ≤ 5 ↔ (𝑀 + 1) ≤ (5 + 1))
11	5, 10	mpbir 231	. . 3 ⊢ 𝑀 ≤ 5
12		6re 12247	. . . 4 ⊢ 6 ∈ ℝ
13		5lt6 12333	. . . 4 ⊢ 5 < 6
14	8, 12, 13	ltleii 11268	. . 3 ⊢ 5 ≤ 6
15	7, 8, 12	letri 11274	. . 3 ⊢ ((𝑀 ≤ 5 ∧ 5 ≤ 6) → 𝑀 ≤ 6)
16	11, 14, 15	mp2an 693	. 2 ⊢ 𝑀 ≤ 6
17	6	nn0zi 12528	. . . . 5 ⊢ 𝑀 ∈ ℤ
18		5nn 12243	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12527	. . . . 5 ⊢ 5 ∈ ℤ
20		eluz2 12769	. . . . 5 ⊢ (5 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 5 ∈ ℤ ∧ 𝑀 ≤ 5))
21	17, 19, 11, 20	mpbir3an 1343	. . . 4 ⊢ 5 ∈ (ℤ_≥‘𝑀)
22		elfzp12 13531	. . . 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 12230	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
26		6nn 12246	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
27		prmz 16614	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
28	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
29		3z 12536	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
30		2z 12535	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
31		dvdsmul2 16217	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (3 · 2))
32	29, 30, 31	mp2an 693	. . . . . . . . . . . . 13 ⊢ 2 ∥ (3 · 2)
33		3t2e6 12318	. . . . . . . . . . . . 13 ⊢ (3 · 2) = 6
34	32, 33	breqtri 5125	. . . . . . . . . . . 12 ⊢ 2 ∥ 6
35		dvdsmod 16268	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 2 ∥ 6) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
36	34, 35	mpan2 692	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
37	25, 26, 28, 36	mp3an12i 1468	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
38		uzid 12778	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
39	30, 38	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
40		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℙ)
41		dvdsprm 16642	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
42	39, 40, 41	sylancr 588	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
43	37, 42	bitrd 279	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 = 𝑃))
44		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 4 ≤ 𝑃)
45		breq2 5104	. . . . . . . . . . 11 ⊢ (2 = 𝑃 → (4 ≤ 2 ↔ 4 ≤ 𝑃))
46	44, 45	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → 4 ≤ 2))
47		2lt4 12327	. . . . . . . . . . . 12 ⊢ 2 < 4
48		2re 12231	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
49		4re 12241	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
50	48, 49	ltnlei 11266	. . . . . . . . . . . 12 ⊢ (2 < 4 ↔ ¬ 4 ≤ 2)
51	47, 50	mpbi 230	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 2
52	51	pm2.21i 119	. . . . . . . . . 10 ⊢ (4 ≤ 2 → (𝑃 mod 6) ∈ {1, 5})
53	46, 52	syl6 35	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
54	43, 53	sylbid 240	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
55		breq2 5104	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑀))
56	55	imbi1d 341	. . . . . . . 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 245	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (2 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
58	57	com3r 87	. . . . . 6 ⊢ (2 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
59		3nn 12236	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
60		dvdsmul1 16216	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
61	29, 30, 60	mp2an 693	. . . . . . . . . . . . 13 ⊢ 3 ∥ (3 · 2)
62	61, 33	breqtri 5125	. . . . . . . . . . . 12 ⊢ 3 ∥ 6
63		dvdsmod 16268	. . . . . . . . . . . 12 ⊢ (((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 3 ∥ 6) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
64	62, 63	mpan2 692	. . . . . . . . . . 11 ⊢ ((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
65	59, 26, 28, 64	mp3an12i 1468	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
66		df-3 12221	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
67		peano2uz 12826	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘2) → (2 + 1) ∈ (ℤ_≥‘2))
68	39, 67	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2 + 1) ∈ (ℤ_≥‘2)
69	66, 68	eqeltri 2833	. . . . . . . . . . 11 ⊢ 3 ∈ (ℤ_≥‘2)
70		dvdsprm 16642	. . . . . . . . . . 11 ⊢ ((3 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
71	69, 40, 70	sylancr 588	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
72	65, 71	bitrd 279	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 = 𝑃))
73		breq2 5104	. . . . . . . . . . 11 ⊢ (3 = 𝑃 → (4 ≤ 3 ↔ 4 ≤ 𝑃))
74	44, 73	syl5ibrcom 247	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → 4 ≤ 3))
75		3lt4 12326	. . . . . . . . . . . 12 ⊢ 3 < 4
76		3re 12237	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
77	76, 49	ltnlei 11266	. . . . . . . . . . . 12 ⊢ (3 < 4 ↔ ¬ 4 ≤ 3)
78	75, 77	mpbi 230	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 3
79	78	pm2.21i 119	. . . . . . . . . 10 ⊢ (4 ≤ 3 → (𝑃 mod 6) ∈ {1, 5})
80	74, 79	syl6 35	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
81	72, 80	sylbid 240	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
82		breq2 5104	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑀))
83	82	imbi1d 341	. . . . . . . 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 245	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (3 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
85	84	com3r 87	. . . . . 6 ⊢ (3 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
86		eleq1a 2832	. . . . . . 7 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5}))
87	86	a1d 25	. . . . . 6 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
88	58, 85, 87	3jaoi 1431	. . . . 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 7378	. . . . . 6 ⊢ (𝑁...5) = ((𝑀 + 1)...5)
91	90	eleq2i 2829	. . . . 5 ⊢ ((𝑃 mod 6) ∈ (𝑁...5) ↔ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))
92	1	simpri 485	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5}))
93	91, 92	biimtrrid 243	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ ((𝑀 + 1)...5) → (𝑃 mod 6) ∈ {1, 5}))
94	89, 93	jaod 860	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5)) → (𝑃 mod 6) ∈ {1, 5}))
95	23, 94	biimtrid 242	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5}))
96	16, 95	pm3.2i 470	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 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cpr 4584 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 ℕcn 12157 2c2 12212 3c3 12213 4c4 12214 5c5 12215 6c6 12216 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 mod cmo 13801 ∥ cdvds 16191 ℙcprime 16610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-dvds 16192 df-prm 16611

This theorem is referenced by: ppiublem2 27182

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