ppiublem1 - Metamath Proof Explorer

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Theorem ppiublem1 26694

Description: Lemma for ppiub 26696. (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 484	. . . . 5 ⊢ 𝑁 ≤ 6
3		ppiublem1.3	. . . . 5 ⊢ 𝑁 = (𝑀 + 1)
4		df-6 12275	. . . . 5 ⊢ 6 = (5 + 1)
5	2, 3, 4	3brtr3i 5176	. . . 4 ⊢ (𝑀 + 1) ≤ (5 + 1)
6		ppiublem1.2	. . . . . 6 ⊢ 𝑀 ∈ ℕ₀
7	6	nn0rei 12479	. . . . 5 ⊢ 𝑀 ∈ ℝ
8		5re 12295	. . . . 5 ⊢ 5 ∈ ℝ
9		1re 11210	. . . . 5 ⊢ 1 ∈ ℝ
10	7, 8, 9	leadd1i 11765	. . . 4 ⊢ (𝑀 ≤ 5 ↔ (𝑀 + 1) ≤ (5 + 1))
11	5, 10	mpbir 230	. . 3 ⊢ 𝑀 ≤ 5
12		6re 12298	. . . 4 ⊢ 6 ∈ ℝ
13		5lt6 12389	. . . 4 ⊢ 5 < 6
14	8, 12, 13	ltleii 11333	. . 3 ⊢ 5 ≤ 6
15	7, 8, 12	letri 11339	. . 3 ⊢ ((𝑀 ≤ 5 ∧ 5 ≤ 6) → 𝑀 ≤ 6)
16	11, 14, 15	mp2an 690	. 2 ⊢ 𝑀 ≤ 6
17	6	nn0zi 12583	. . . . 5 ⊢ 𝑀 ∈ ℤ
18		5nn 12294	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12582	. . . . 5 ⊢ 5 ∈ ℤ
20		eluz2 12824	. . . . 5 ⊢ (5 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 5 ∈ ℤ ∧ 𝑀 ≤ 5))
21	17, 19, 11, 20	mpbir3an 1341	. . . 4 ⊢ 5 ∈ (ℤ_≥‘𝑀)
22		elfzp12 13576	. . . 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 12281	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
26		6nn 12297	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
27		prmz 16608	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
28	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
29		3z 12591	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
30		2z 12590	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
31		dvdsmul2 16218	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (3 · 2))
32	29, 30, 31	mp2an 690	. . . . . . . . . . . . 13 ⊢ 2 ∥ (3 · 2)
33		3t2e6 12374	. . . . . . . . . . . . 13 ⊢ (3 · 2) = 6
34	32, 33	breqtri 5172	. . . . . . . . . . . 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 689	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
37	25, 26, 28, 36	mp3an12i 1465	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
38		uzid 12833	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
39	30, 38	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
40		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℙ)
41		dvdsprm 16636	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
42	39, 40, 41	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
43	37, 42	bitrd 278	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 = 𝑃))
44		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 4 ≤ 𝑃)
45		breq2 5151	. . . . . . . . . . 11 ⊢ (2 = 𝑃 → (4 ≤ 2 ↔ 4 ≤ 𝑃))
46	44, 45	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → 4 ≤ 2))
47		2lt4 12383	. . . . . . . . . . . 12 ⊢ 2 < 4
48		2re 12282	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
49		4re 12292	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
50	48, 49	ltnlei 11331	. . . . . . . . . . . 12 ⊢ (2 < 4 ↔ ¬ 4 ≤ 2)
51	47, 50	mpbi 229	. . . . . . . . . . 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 239	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
55		breq2 5151	. . . . . . . . 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 244	. . . . . . 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 12287	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
60		dvdsmul1 16217	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
61	29, 30, 60	mp2an 690	. . . . . . . . . . . . 13 ⊢ 3 ∥ (3 · 2)
62	61, 33	breqtri 5172	. . . . . . . . . . . 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 689	. . . . . . . . . . 11 ⊢ ((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
65	59, 26, 28, 64	mp3an12i 1465	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
66		df-3 12272	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
67		peano2uz 12881	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘2) → (2 + 1) ∈ (ℤ_≥‘2))
68	39, 67	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2 + 1) ∈ (ℤ_≥‘2)
69	66, 68	eqeltri 2829	. . . . . . . . . . 11 ⊢ 3 ∈ (ℤ_≥‘2)
70		dvdsprm 16636	. . . . . . . . . . 11 ⊢ ((3 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
71	69, 40, 70	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
72	65, 71	bitrd 278	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 = 𝑃))
73		breq2 5151	. . . . . . . . . . 11 ⊢ (3 = 𝑃 → (4 ≤ 3 ↔ 4 ≤ 𝑃))
74	44, 73	syl5ibrcom 246	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → 4 ≤ 3))
75		3lt4 12382	. . . . . . . . . . . 12 ⊢ 3 < 4
76		3re 12288	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
77	76, 49	ltnlei 11331	. . . . . . . . . . . 12 ⊢ (3 < 4 ↔ ¬ 4 ≤ 3)
78	75, 77	mpbi 229	. . . . . . . . . . 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 239	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
82		breq2 5151	. . . . . . . . 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 244	. . . . . . 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 2828	. . . . . . 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 1427	. . . . 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 7415	. . . . . 6 ⊢ (𝑁...5) = ((𝑀 + 1)...5)
91	90	eleq2i 2825	. . . . 5 ⊢ ((𝑃 mod 6) ∈ (𝑁...5) ↔ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))
92	1	simpri 486	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5}))
93	91, 92	biimtrrid 242	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ ((𝑀 + 1)...5) → (𝑃 mod 6) ∈ {1, 5}))
94	89, 93	jaod 857	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5)) → (𝑃 mod 6) ∈ {1, 5}))
95	23, 94	biimtrid 241	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5}))
96	16, 95	pm3.2i 471	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 205 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cpr 4629 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 ℕcn 12208 2c2 12263 3c3 12264 4c4 12265 5c5 12266 6c6 12267 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 mod cmo 13830 ∥ cdvds 16193 ℙcprime 16604

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-prm 16605

This theorem is referenced by: ppiublem2 26695

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