ppiublem1 - Metamath Proof Explorer

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Theorem ppiublem1 24927

Description: Lemma for ppiub 24929. (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 474	. . . . 5 ⊢ 𝑁 ≤ 6
3		ppiublem1.3	. . . . 5 ⊢ 𝑁 = (𝑀 + 1)
4		df-6 11083	. . . . 5 ⊢ 6 = (5 + 1)
5	2, 3, 4	3brtr3i 4682	. . . 4 ⊢ (𝑀 + 1) ≤ (5 + 1)
6		ppiublem1.2	. . . . . 6 ⊢ 𝑀 ∈ ℕ₀
7	6	nn0rei 11303	. . . . 5 ⊢ 𝑀 ∈ ℝ
8		5re 11099	. . . . 5 ⊢ 5 ∈ ℝ
9		1re 10039	. . . . 5 ⊢ 1 ∈ ℝ
10	7, 8, 9	leadd1i 10583	. . . 4 ⊢ (𝑀 ≤ 5 ↔ (𝑀 + 1) ≤ (5 + 1))
11	5, 10	mpbir 221	. . 3 ⊢ 𝑀 ≤ 5
12		6re 11101	. . . 4 ⊢ 6 ∈ ℝ
13		5lt6 11204	. . . 4 ⊢ 5 < 6
14	8, 12, 13	ltleii 10160	. . 3 ⊢ 5 ≤ 6
15	7, 8, 12	letri 10166	. . 3 ⊢ ((𝑀 ≤ 5 ∧ 5 ≤ 6) → 𝑀 ≤ 6)
16	11, 14, 15	mp2an 708	. 2 ⊢ 𝑀 ≤ 6
17	6	nn0zi 11402	. . . . 5 ⊢ 𝑀 ∈ ℤ
18		5nn 11188	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 11401	. . . . 5 ⊢ 5 ∈ ℤ
20		eluz2 11693	. . . . 5 ⊢ (5 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 5 ∈ ℤ ∧ 𝑀 ≤ 5))
21	17, 19, 11, 20	mpbir3an 1244	. . . 4 ⊢ 5 ∈ (ℤ_≥‘𝑀)
22		elfzp12 12419	. . . 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		prmz 15389	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
27		2nn 11185	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
28		6nn 11189	. . . . . . . . . . . 12 ⊢ 6 ∈ ℕ
29		3z 11410	. . . . . . . . . . . . . . 15 ⊢ 3 ∈ ℤ
30		2z 11409	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℤ
31		dvdsmul2 15004	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (3 · 2))
32	29, 30, 31	mp2an 708	. . . . . . . . . . . . . 14 ⊢ 2 ∥ (3 · 2)
33		3t2e6 11179	. . . . . . . . . . . . . 14 ⊢ (3 · 2) = 6
34	32, 33	breqtri 4678	. . . . . . . . . . . . 13 ⊢ 2 ∥ 6
35		dvdsmod 15050	. . . . . . . . . . . . 13 ⊢ (((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 2 ∥ 6) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
36	34, 35	mpan2 707	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
37	27, 28, 36	mp3an12 1414	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℤ → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
38	26, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
39		uzid 11702	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
40	30, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
41		simpl 473	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℙ)
42		dvdsprm 15415	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
43	40, 41, 42	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
44	38, 43	bitrd 268	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 = 𝑃))
45		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 4 ≤ 𝑃)
46		breq2 4657	. . . . . . . . . . 11 ⊢ (2 = 𝑃 → (4 ≤ 2 ↔ 4 ≤ 𝑃))
47	45, 46	syl5ibrcom 237	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → 4 ≤ 2))
48		2lt4 11198	. . . . . . . . . . . 12 ⊢ 2 < 4
49		2re 11090	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
50		4re 11097	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
51	49, 50	ltnlei 10158	. . . . . . . . . . . 12 ⊢ (2 < 4 ↔ ¬ 4 ≤ 2)
52	48, 51	mpbi 220	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 2
53	52	pm2.21i 116	. . . . . . . . . 10 ⊢ (4 ≤ 2 → (𝑃 mod 6) ∈ {1, 5})
54	47, 53	syl6 35	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
55	44, 54	sylbid 230	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
56		breq2 4657	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑀))
57	56	imbi1d 331	. . . . . . . 8 ⊢ ((𝑃 mod 6) = 𝑀 → ((2 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}) ↔ (2 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
58	55, 57	syl5ibcom 235	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (2 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
59	58	com3r 87	. . . . . 6 ⊢ (2 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
60		3nn 11186	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ
61		dvdsmul1 15003	. . . . . . . . . . . . . . 15 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
62	29, 30, 61	mp2an 708	. . . . . . . . . . . . . 14 ⊢ 3 ∥ (3 · 2)
63	62, 33	breqtri 4678	. . . . . . . . . . . . 13 ⊢ 3 ∥ 6
64		dvdsmod 15050	. . . . . . . . . . . . 13 ⊢ (((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 3 ∥ 6) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
65	63, 64	mpan2 707	. . . . . . . . . . . 12 ⊢ ((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
66	60, 28, 65	mp3an12 1414	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℤ → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
67	26, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
68		df-3 11080	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
69		peano2uz 11741	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘2) → (2 + 1) ∈ (ℤ_≥‘2))
70	40, 69	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2 + 1) ∈ (ℤ_≥‘2)
71	68, 70	eqeltri 2697	. . . . . . . . . . 11 ⊢ 3 ∈ (ℤ_≥‘2)
72		dvdsprm 15415	. . . . . . . . . . 11 ⊢ ((3 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
73	71, 41, 72	sylancr 695	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
74	67, 73	bitrd 268	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 = 𝑃))
75		breq2 4657	. . . . . . . . . . 11 ⊢ (3 = 𝑃 → (4 ≤ 3 ↔ 4 ≤ 𝑃))
76	45, 75	syl5ibrcom 237	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → 4 ≤ 3))
77		3lt4 11197	. . . . . . . . . . . 12 ⊢ 3 < 4
78		3re 11094	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
79	78, 50	ltnlei 10158	. . . . . . . . . . . 12 ⊢ (3 < 4 ↔ ¬ 4 ≤ 3)
80	77, 79	mpbi 220	. . . . . . . . . . 11 ⊢ ¬ 4 ≤ 3
81	80	pm2.21i 116	. . . . . . . . . 10 ⊢ (4 ≤ 3 → (𝑃 mod 6) ∈ {1, 5})
82	76, 81	syl6 35	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → (𝑃 mod 6) ∈ {1, 5}))
83	74, 82	sylbid 230	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}))
84		breq2 4657	. . . . . . . . 9 ⊢ ((𝑃 mod 6) = 𝑀 → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑀))
85	84	imbi1d 331	. . . . . . . 8 ⊢ ((𝑃 mod 6) = 𝑀 → ((3 ∥ (𝑃 mod 6) → (𝑃 mod 6) ∈ {1, 5}) ↔ (3 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
86	83, 85	syl5ibcom 235	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (3 ∥ 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
87	86	com3r 87	. . . . . 6 ⊢ (3 ∥ 𝑀 → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
88		eleq1a 2696	. . . . . . 7 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5}))
89	88	a1d 25	. . . . . 6 ⊢ (𝑀 ∈ {1, 5} → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
90	59, 87, 89	3jaoi 1391	. . . . 5 ⊢ ((2 ∥ 𝑀 ∨ 3 ∥ 𝑀 ∨ 𝑀 ∈ {1, 5}) → ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5})))
91	24, 90	ax-mp 5	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) = 𝑀 → (𝑃 mod 6) ∈ {1, 5}))
92	3	oveq1i 6660	. . . . . 6 ⊢ (𝑁...5) = ((𝑀 + 1)...5)
93	92	eleq2i 2693	. . . . 5 ⊢ ((𝑃 mod 6) ∈ (𝑁...5) ↔ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))
94	1	simpri 478	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑁...5) → (𝑃 mod 6) ∈ {1, 5}))
95	93, 94	syl5bir 233	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ ((𝑀 + 1)...5) → (𝑃 mod 6) ∈ {1, 5}))
96	91, 95	jaod 395	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (((𝑃 mod 6) = 𝑀 ∨ (𝑃 mod 6) ∈ ((𝑀 + 1)...5)) → (𝑃 mod 6) ∈ {1, 5}))
97	23, 96	syl5bi 232	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5}))
98	16, 97	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 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cpr 4179 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 5c5 11073 6c6 11074 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 mod cmo 12668 ∥ cdvds 14983 ℙcprime 15385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386

This theorem is referenced by: ppiublem2 24928

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