ppiublem1 - Metamath Proof Explorer

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Theorem ppiublem1 27536

Description: Lemma for ppiub 27538. (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 489	. . . . 5 ⊢ 𝑁 ≤ 6
3		ppiublem1.3	. . . . 5 ⊢ 𝑁 = (𝑀 + 1)
4		df-6 12414	. . . . 5 ⊢ 6 = (5 + 1)
5	2, 3, 4	3brtr3i 5133	. . . 4 ⊢ (𝑀 + 1) ≤ (5 + 1)
6		ppiublem1.2	. . . . . 6 ⊢ 𝑀 ∈ ℕ₀
7	6	nn0rei 12622	. . . . 5 ⊢ 𝑀 ∈ ℝ
8		5re 12435	. . . . 5 ⊢ 5 ∈ ℝ
9		1re 11313	. . . . 5 ⊢ 1 ∈ ℝ
10	7, 8, 9	leadd1i 11876	. . . 4 ⊢ (𝑀 ≤ 5 ↔ (𝑀 + 1) ≤ (5 + 1))
11	5, 10	mpbir 234	. . 3 ⊢ 𝑀 ≤ 5
12		6re 12438	. . . 4 ⊢ 6 ∈ ℝ
13		5lt6 12531	. . . 4 ⊢ 5 < 6
14	8, 12, 13	ltleii 11438	. . 3 ⊢ 5 ≤ 6
15	7, 8, 12	letri 11444	. . 3 ⊢ ((𝑀 ≤ 5 ∧ 5 ≤ 6) → 𝑀 ≤ 6)
16	11, 14, 15	mp2an 705	. 2 ⊢ 𝑀 ≤ 6
17	6	nn0zi 12726	. . . . 5 ⊢ 𝑀 ∈ ℤ
18		5nn 12434	. . . . . 6 ⊢ 5 ∈ ℕ
19	18	nnzi 12725	. . . . 5 ⊢ 5 ∈ ℤ
20		eluz2 12976	. . . . 5 ⊢ (5 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 5 ∈ ℤ ∧ 𝑀 ≤ 5))
21	17, 19, 11, 20	mpbir3an 1360	. . . 4 ⊢ 5 ∈ (ℤ_≥‘𝑀)
22		elfzp12 13742	. . . 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 12421	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ
26		6nn 12437	. . . . . . . . . . 11 ⊢ 6 ∈ ℕ
27		prmz 16857	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
28	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℤ)
29		3z 12734	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
30		2z 12733	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
31		dvdsmul2 16455	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (3 · 2))
32	29, 30, 31	mp2an 705	. . . . . . . . . . . . 13 ⊢ 2 ∥ (3 · 2)
33		3t2e6 12513	. . . . . . . . . . . . 13 ⊢ (3 · 2) = 6
34	32, 33	breqtri 5129	. . . . . . . . . . . 12 ⊢ 2 ∥ 6
35		dvdsmod 16506	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 2 ∥ 6) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
36	34, 35	mpan2 704	. . . . . . . . . . 11 ⊢ ((2 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
37	25, 26, 28, 36	mp3an12i 1494	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 ∥ (𝑃 mod 6) ↔ 2 ∥ 𝑃))
38		uzid 12985	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
39	30, 38	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
40		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 𝑃 ∈ ℙ)
41		dvdsprm 16886	. . . . . . . . . . 11 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (2 ∥ 𝑃 ↔ 2 = 𝑃))
42	39, 40, 41	sylancr 599	. . . . . . . . . 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 490	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → 4 ≤ 𝑃)
45		breq2 5106	. . . . . . . . . . 11 ⊢ (2 = 𝑃 → (4 ≤ 2 ↔ 4 ≤ 𝑃))
46	44, 45	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (2 = 𝑃 → 4 ≤ 2))
47		2lt4 12525	. . . . . . . . . . . 12 ⊢ 2 < 4
48		2re 12422	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
49		4re 12432	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℝ
50	48, 49	ltnlei 11436	. . . . . . . . . . . 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 5106	. . . . . . . . 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 12427	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ
60		dvdsmul1 16454	. . . . . . . . . . . . . 14 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ) → 3 ∥ (3 · 2))
61	29, 30, 60	mp2an 705	. . . . . . . . . . . . 13 ⊢ 3 ∥ (3 · 2)
62	61, 33	breqtri 5129	. . . . . . . . . . . 12 ⊢ 3 ∥ 6
63		dvdsmod 16506	. . . . . . . . . . . 12 ⊢ (((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) ∧ 3 ∥ 6) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
64	62, 63	mpan2 704	. . . . . . . . . . 11 ⊢ ((3 ∈ ℕ ∧ 6 ∈ ℕ ∧ 𝑃 ∈ ℤ) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
65	59, 26, 28, 64	mp3an12i 1494	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 ∥ (𝑃 mod 6) ↔ 3 ∥ 𝑃))
66		df-3 12411	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
67		peano2uz 13033	. . . . . . . . . . . . 13 ⊢ (2 ∈ (ℤ_≥‘2) → (2 + 1) ∈ (ℤ_≥‘2))
68	39, 67	ax-mp 5	. . . . . . . . . . . 12 ⊢ (2 + 1) ∈ (ℤ_≥‘2)
69	66, 68	eqeltri 2856	. . . . . . . . . . 11 ⊢ 3 ∈ (ℤ_≥‘2)
70		dvdsprm 16886	. . . . . . . . . . 11 ⊢ ((3 ∈ (ℤ_≥‘2) ∧ 𝑃 ∈ ℙ) → (3 ∥ 𝑃 ↔ 3 = 𝑃))
71	69, 40, 70	sylancr 599	. . . . . . . . . 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 5106	. . . . . . . . . . 11 ⊢ (3 = 𝑃 → (4 ≤ 3 ↔ 4 ≤ 𝑃))
74	44, 73	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → (3 = 𝑃 → 4 ≤ 3))
75		3lt4 12524	. . . . . . . . . . . 12 ⊢ 3 < 4
76		3re 12428	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℝ
77	76, 49	ltnlei 11436	. . . . . . . . . . . 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 5106	. . . . . . . . 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 2855	. . . . . . 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 1454	. . . . 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 7432	. . . . . 6 ⊢ (𝑁...5) = ((𝑀 + 1)...5)
91	90	eleq2i 2852	. . . . 5 ⊢ ((𝑃 mod 6) ∈ (𝑁...5) ↔ (𝑃 mod 6) ∈ ((𝑀 + 1)...5))
92	1	simpri 491	. . . . 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 873	. . 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 476	1 ⊢ (𝑀 ≤ 6 ∧ ((𝑃 ∈ ℙ ∧ 4 ≤ 𝑃) → ((𝑃 mod 6) ∈ (𝑀...5) → (𝑃 mod 6) ∈ {1, 5})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 {cpr 4586 class class class wbr 5102 ‘cfv 6541 (class class class)co 7422 1c1 11206 + caddc 11208 · cmul 11210 < clt 11348 ≤ cle 11349 ℕcn 12340 2c2 12402 3c3 12403 4c4 12404 5c5 12405 6c6 12406 ℕ₀cn0 12611 ℤcz 12698 ℤ_≥cuz 12970 ...cfz 13644 mod cmo 14014 ∥ cdvds 16429 ℙcprime 16853

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282 ax-pre-sup 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-2o 8482 df-er 8722 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-sup 9439 df-inf 9440 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-div 11979 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-n0 12612 df-z 12699 df-uz 12971 df-rp 13126 df-fz 13645 df-fl 13937 df-mod 14015 df-seq 14150 df-exp 14210 df-cj 15273 df-re 15274 df-im 15275 df-sqrt 15409 df-abs 15410 df-dvds 16430 df-prm 16854

This theorem is used by: ppiublem2 27537

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