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Theorem pockthlem 16952

Description: Lemma for pockthg 16953. (Contributed by Mario Carneiro, 2-Mar-2014.)

**Hypotheses**
Ref	Expression
pockthg.1	⊢ (𝜑 → 𝐴 ∈ ℕ)
pockthg.2	⊢ (𝜑 → 𝐵 ∈ ℕ)
pockthg.3	⊢ (𝜑 → 𝐵 < 𝐴)
pockthg.4	⊢ (𝜑 → 𝑁 = ((𝐴 · 𝐵) + 1))
pockthlem.5	⊢ (𝜑 → 𝑃 ∈ ℙ)
pockthlem.6	⊢ (𝜑 → 𝑃 ∥ 𝑁)
pockthlem.7	⊢ (𝜑 → 𝑄 ∈ ℙ)
pockthlem.8	⊢ (𝜑 → (𝑄 pCnt 𝐴) ∈ ℕ)
pockthlem.9	⊢ (𝜑 → 𝐶 ∈ ℤ)
pockthlem.10	⊢ (𝜑 → ((𝐶↑(𝑁 − 1)) mod 𝑁) = 1)
pockthlem.11	⊢ (𝜑 → (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁) = 1)

**Assertion**
Ref	Expression
pockthlem	⊢ (𝜑 → (𝑄 pCnt 𝐴) ≤ (𝑄 pCnt (𝑃 − 1)))

**Proof of Theorem pockthlem**
Step	Hyp	Ref	Expression
1		pockthlem.7	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℙ)
2		prmnn 16721	. . . . . 6 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ ℕ)
4		pockthlem.8	. . . . . 6 ⊢ (𝜑 → (𝑄 pCnt 𝐴) ∈ ℕ)
5	4	nnnn0d 12613	. . . . 5 ⊢ (𝜑 → (𝑄 pCnt 𝐴) ∈ ℕ₀)
6	3, 5	nnexpcld 14294	. . . 4 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∈ ℕ)
7	6	nnzd 12666	. . 3 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∈ ℤ)
8		pockthlem.5	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
9		prmnn 16721	. . . . . 6 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℕ)
11		pockthlem.9	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ)
12	10	nnzd 12666	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ℤ)
13		gcddvds 16549	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝐶 gcd 𝑃) ∥ 𝐶 ∧ (𝐶 gcd 𝑃) ∥ 𝑃))
14	11, 12, 13	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 gcd 𝑃) ∥ 𝐶 ∧ (𝐶 gcd 𝑃) ∥ 𝑃))
15	14	simpld 494	. . . . . . . 8 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∥ 𝐶)
16	11, 12	gcdcld 16554	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∈ ℕ₀)
17	16	nn0zd 12665	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∈ ℤ)
18		pockthg.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 = ((𝐴 · 𝐵) + 1))
19		pockthg.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ∈ ℕ)
20		pockthg.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℕ)
21	19, 20	nnmulcld 12346	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℕ)
22		nnuz 12946	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
23	21, 22	eleqtrdi 2854	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ (ℤ_≥‘1))
24		eluzp1p1 12931	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 · 𝐵) ∈ (ℤ_≥‘1) → ((𝐴 · 𝐵) + 1) ∈ (ℤ_≥‘(1 + 1)))
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐴 · 𝐵) + 1) ∈ (ℤ_≥‘(1 + 1)))
26	18, 25	eqeltrd 2844	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(1 + 1)))
27		df-2 12356	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
28	27	fveq2i 6923	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
29	26, 28	eleqtrrdi 2855	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))
30		eluz2b2 12986	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
31	29, 30	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
32	31	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
33	32	nnzd 12666	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
34	14	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∥ 𝑃)
35		pockthlem.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
36	17, 12, 33, 34, 35	dvdstrd 16343	. . . . . . . 8 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∥ 𝑁)
37	32	nnne0d 12343	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ≠ 0)
38		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐶 = 0 ∧ 𝑁 = 0) → 𝑁 = 0)
39	38	necon3ai 2971	. . . . . . . . . 10 ⊢ (𝑁 ≠ 0 → ¬ (𝐶 = 0 ∧ 𝑁 = 0))
40	37, 39	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ¬ (𝐶 = 0 ∧ 𝑁 = 0))
41		dvdslegcd 16550	. . . . . . . . 9 ⊢ ((((𝐶 gcd 𝑃) ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝑁 = 0)) → (((𝐶 gcd 𝑃) ∥ 𝐶 ∧ (𝐶 gcd 𝑃) ∥ 𝑁) → (𝐶 gcd 𝑃) ≤ (𝐶 gcd 𝑁)))
42	17, 11, 33, 40, 41	syl31anc 1373	. . . . . . . 8 ⊢ (𝜑 → (((𝐶 gcd 𝑃) ∥ 𝐶 ∧ (𝐶 gcd 𝑃) ∥ 𝑁) → (𝐶 gcd 𝑃) ≤ (𝐶 gcd 𝑁)))
43	15, 36, 42	mp2and 698	. . . . . . 7 ⊢ (𝜑 → (𝐶 gcd 𝑃) ≤ (𝐶 gcd 𝑁))
44		pockthlem.10	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐶↑(𝑁 − 1)) mod 𝑁) = 1)
45	44	oveq1d 7463	. . . . . . . . 9 ⊢ (𝜑 → (((𝐶↑(𝑁 − 1)) mod 𝑁) gcd 𝑁) = (1 gcd 𝑁))
46		1z 12673	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
47		eluzp1m1 12929	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ (ℤ_≥‘(1 + 1))) → (𝑁 − 1) ∈ (ℤ_≥‘1))
48	46, 26, 47	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘1))
49	48, 22	eleqtrrdi 2855	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ)
50	49	nnnn0d 12613	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
51		zexpcl 14127	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℤ ∧ (𝑁 − 1) ∈ ℕ₀) → (𝐶↑(𝑁 − 1)) ∈ ℤ)
52	11, 50, 51	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶↑(𝑁 − 1)) ∈ ℤ)
53		modgcd 16579	. . . . . . . . . 10 ⊢ (((𝐶↑(𝑁 − 1)) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐶↑(𝑁 − 1)) mod 𝑁) gcd 𝑁) = ((𝐶↑(𝑁 − 1)) gcd 𝑁))
54	52, 32, 53	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (((𝐶↑(𝑁 − 1)) mod 𝑁) gcd 𝑁) = ((𝐶↑(𝑁 − 1)) gcd 𝑁))
55		gcdcom 16559	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (1 gcd 𝑁) = (𝑁 gcd 1))
56	46, 33, 55	sylancr 586	. . . . . . . . . 10 ⊢ (𝜑 → (1 gcd 𝑁) = (𝑁 gcd 1))
57		gcd1 16574	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑁 gcd 1) = 1)
58	33, 57	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 gcd 1) = 1)
59	56, 58	eqtrd 2780	. . . . . . . . 9 ⊢ (𝜑 → (1 gcd 𝑁) = 1)
60	45, 54, 59	3eqtr3d 2788	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑(𝑁 − 1)) gcd 𝑁) = 1)
61		rpexp 16769	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁 − 1) ∈ ℕ) → (((𝐶↑(𝑁 − 1)) gcd 𝑁) = 1 ↔ (𝐶 gcd 𝑁) = 1))
62	11, 33, 49, 61	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (((𝐶↑(𝑁 − 1)) gcd 𝑁) = 1 ↔ (𝐶 gcd 𝑁) = 1))
63	60, 62	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐶 gcd 𝑁) = 1)
64	43, 63	breqtrd 5192	. . . . . 6 ⊢ (𝜑 → (𝐶 gcd 𝑃) ≤ 1)
65	10	nnne0d 12343	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 0)
66		simpr 484	. . . . . . . . . 10 ⊢ ((𝐶 = 0 ∧ 𝑃 = 0) → 𝑃 = 0)
67	66	necon3ai 2971	. . . . . . . . 9 ⊢ (𝑃 ≠ 0 → ¬ (𝐶 = 0 ∧ 𝑃 = 0))
68	65, 67	syl 17	. . . . . . . 8 ⊢ (𝜑 → ¬ (𝐶 = 0 ∧ 𝑃 = 0))
69		gcdn0cl 16548	. . . . . . . 8 ⊢ (((𝐶 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ ¬ (𝐶 = 0 ∧ 𝑃 = 0)) → (𝐶 gcd 𝑃) ∈ ℕ)
70	11, 12, 68, 69	syl21anc 837	. . . . . . 7 ⊢ (𝜑 → (𝐶 gcd 𝑃) ∈ ℕ)
71		nnle1eq1 12323	. . . . . . 7 ⊢ ((𝐶 gcd 𝑃) ∈ ℕ → ((𝐶 gcd 𝑃) ≤ 1 ↔ (𝐶 gcd 𝑃) = 1))
72	70, 71	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐶 gcd 𝑃) ≤ 1 ↔ (𝐶 gcd 𝑃) = 1))
73	64, 72	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝐶 gcd 𝑃) = 1)
74		odzcl 16840	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝐶 ∈ ℤ ∧ (𝐶 gcd 𝑃) = 1) → ((od_ℤ‘𝑃)‘𝐶) ∈ ℕ)
75	10, 11, 73, 74	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∈ ℕ)
76	75	nnzd 12666	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∈ ℤ)
77		prmuz2 16743	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
78	8, 77	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘2))
79	78, 28	eleqtrdi 2854	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (ℤ_≥‘(1 + 1)))
80		eluzp1m1 12929	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑃 ∈ (ℤ_≥‘(1 + 1))) → (𝑃 − 1) ∈ (ℤ_≥‘1))
81	46, 79, 80	sylancr 586	. . . . 5 ⊢ (𝜑 → (𝑃 − 1) ∈ (ℤ_≥‘1))
82	81, 22	eleqtrrdi 2855	. . . 4 ⊢ (𝜑 → (𝑃 − 1) ∈ ℕ)
83	82	nnzd 12666	. . 3 ⊢ (𝜑 → (𝑃 − 1) ∈ ℤ)
84	19	nnzd 12666	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
85	49	nnzd 12666	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
86		pcdvds 16911	. . . . . . 7 ⊢ ((𝑄 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑄↑(𝑄 pCnt 𝐴)) ∥ 𝐴)
87	1, 19, 86	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∥ 𝐴)
88	20	nnzd 12666	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℤ)
89		dvdsmul1 16326	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∥ (𝐴 · 𝐵))
90	84, 88, 89	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∥ (𝐴 · 𝐵))
91	18	oveq1d 7463	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 1) = (((𝐴 · 𝐵) + 1) − 1))
92	21	nncnd 12309	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 · 𝐵) ∈ ℂ)
93		ax-1cn 11242	. . . . . . . . 9 ⊢ 1 ∈ ℂ
94		pncan 11542	. . . . . . . . 9 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐴 · 𝐵) + 1) − 1) = (𝐴 · 𝐵))
95	92, 93, 94	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 · 𝐵) + 1) − 1) = (𝐴 · 𝐵))
96	91, 95	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → (𝑁 − 1) = (𝐴 · 𝐵))
97	90, 96	breqtrrd 5194	. . . . . 6 ⊢ (𝜑 → 𝐴 ∥ (𝑁 − 1))
98	7, 84, 85, 87, 97	dvdstrd 16343	. . . . 5 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑁 − 1))
99	6	nnne0d 12343	. . . . . 6 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ≠ 0)
100		dvdsval2 16305	. . . . . 6 ⊢ (((𝑄↑(𝑄 pCnt 𝐴)) ∈ ℤ ∧ (𝑄↑(𝑄 pCnt 𝐴)) ≠ 0 ∧ (𝑁 − 1) ∈ ℤ) → ((𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℤ))
101	7, 99, 85, 100	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℤ))
102	98, 101	mpbid 232	. . . 4 ⊢ (𝜑 → ((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℤ)
103		peano2zm 12686	. . . . . . . 8 ⊢ ((𝐶↑(𝑁 − 1)) ∈ ℤ → ((𝐶↑(𝑁 − 1)) − 1) ∈ ℤ)
104	52, 103	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐶↑(𝑁 − 1)) − 1) ∈ ℤ)
105	32	nnred 12308	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
106	31	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 1 < 𝑁)
107		1mod 13954	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 1 < 𝑁) → (1 mod 𝑁) = 1)
108	105, 106, 107	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (1 mod 𝑁) = 1)
109	44, 108	eqtr4d 2783	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁))
110		1zzd 12674	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
111		moddvds 16313	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶↑(𝑁 − 1)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐶↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐶↑(𝑁 − 1)) − 1)))
112	32, 52, 110, 111	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (((𝐶↑(𝑁 − 1)) mod 𝑁) = (1 mod 𝑁) ↔ 𝑁 ∥ ((𝐶↑(𝑁 − 1)) − 1)))
113	109, 112	mpbid 232	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∥ ((𝐶↑(𝑁 − 1)) − 1))
114	12, 33, 104, 35, 113	dvdstrd 16343	. . . . . 6 ⊢ (𝜑 → 𝑃 ∥ ((𝐶↑(𝑁 − 1)) − 1))
115		odzdvds 16842	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ ∧ 𝐶 ∈ ℤ ∧ (𝐶 gcd 𝑃) = 1) ∧ (𝑁 − 1) ∈ ℕ₀) → (𝑃 ∥ ((𝐶↑(𝑁 − 1)) − 1) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ (𝑁 − 1)))
116	10, 11, 73, 50, 115	syl31anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ ((𝐶↑(𝑁 − 1)) − 1) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ (𝑁 − 1)))
117	114, 116	mpbid 232	. . . . 5 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∥ (𝑁 − 1))
118	49	nncnd 12309	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℂ)
119	6	nncnd 12309	. . . . . 6 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∈ ℂ)
120	118, 119, 99	divcan1d 12071	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑(𝑄 pCnt 𝐴))) = (𝑁 − 1))
121	117, 120	breqtrrd 5194	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑(𝑄 pCnt 𝐴))))
122		nprmdvds1 16753	. . . . . 6 ⊢ (𝑃 ∈ ℙ → ¬ 𝑃 ∥ 1)
123	8, 122	syl 17	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ∥ 1)
124	3	nnzd 12666	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℤ)
125		iddvdsexp 16328	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℤ ∧ (𝑄 pCnt 𝐴) ∈ ℕ) → 𝑄 ∥ (𝑄↑(𝑄 pCnt 𝐴)))
126	124, 4, 125	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∥ (𝑄↑(𝑄 pCnt 𝐴)))
127	124, 7, 85, 126, 98	dvdstrd 16343	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∥ (𝑁 − 1))
128	3	nnne0d 12343	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ≠ 0)
129		dvdsval2 16305	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℤ ∧ 𝑄 ≠ 0 ∧ (𝑁 − 1) ∈ ℤ) → (𝑄 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 𝑄) ∈ ℤ))
130	124, 128, 85, 129	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∥ (𝑁 − 1) ↔ ((𝑁 − 1) / 𝑄) ∈ ℤ))
131	127, 130	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) / 𝑄) ∈ ℤ)
132	50	nn0ge0d 12616	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (𝑁 − 1))
133	49	nnred 12308	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
134	3	nnred 12308	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℝ)
135	3	nngt0d 12342	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑄)
136		ge0div 12162	. . . . . . . . . . . . 13 ⊢ (((𝑁 − 1) ∈ ℝ ∧ 𝑄 ∈ ℝ ∧ 0 < 𝑄) → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 𝑄)))
137	133, 134, 135, 136	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (0 ≤ (𝑁 − 1) ↔ 0 ≤ ((𝑁 − 1) / 𝑄)))
138	132, 137	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ ((𝑁 − 1) / 𝑄))
139		elnn0z 12652	. . . . . . . . . . 11 ⊢ (((𝑁 − 1) / 𝑄) ∈ ℕ₀ ↔ (((𝑁 − 1) / 𝑄) ∈ ℤ ∧ 0 ≤ ((𝑁 − 1) / 𝑄)))
140	131, 138, 139	sylanbrc 582	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 − 1) / 𝑄) ∈ ℕ₀)
141		zexpcl 14127	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℤ ∧ ((𝑁 − 1) / 𝑄) ∈ ℕ₀) → (𝐶↑((𝑁 − 1) / 𝑄)) ∈ ℤ)
142	11, 140, 141	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐶↑((𝑁 − 1) / 𝑄)) ∈ ℤ)
143		peano2zm 12686	. . . . . . . . 9 ⊢ ((𝐶↑((𝑁 − 1) / 𝑄)) ∈ ℤ → ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ∈ ℤ)
144	142, 143	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ∈ ℤ)
145		dvdsgcd 16591	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ∧ 𝑃 ∥ 𝑁) → 𝑃 ∥ (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁)))
146	12, 144, 33, 145	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ∧ 𝑃 ∥ 𝑁) → 𝑃 ∥ (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁)))
147	35, 146	mpan2d 693	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) → 𝑃 ∥ (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁)))
148		odzdvds 16842	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ ∧ 𝐶 ∈ ℤ ∧ (𝐶 gcd 𝑃) = 1) ∧ ((𝑁 − 1) / 𝑄) ∈ ℕ₀) → (𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ ((𝑁 − 1) / 𝑄)))
149	10, 11, 73, 140, 148	syl31anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ ((𝑁 − 1) / 𝑄)))
150	3	nncnd 12309	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℂ)
151	4	nnzd 12666	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 pCnt 𝐴) ∈ ℤ)
152	150, 128, 151	expm1d 14206	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄↑((𝑄 pCnt 𝐴) − 1)) = ((𝑄↑(𝑄 pCnt 𝐴)) / 𝑄))
153	152	oveq2d 7464	. . . . . . . . 9 ⊢ (𝜑 → (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))) = (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · ((𝑄↑(𝑄 pCnt 𝐴)) / 𝑄)))
154	133, 6	nndivred 12347	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℝ)
155	154	recnd 11318	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℂ)
156	155, 119, 150, 128	divassd 12105	. . . . . . . . 9 ⊢ (𝜑 → ((((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑(𝑄 pCnt 𝐴))) / 𝑄) = (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · ((𝑄↑(𝑄 pCnt 𝐴)) / 𝑄)))
157	120	oveq1d 7463	. . . . . . . . 9 ⊢ (𝜑 → ((((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑(𝑄 pCnt 𝐴))) / 𝑄) = ((𝑁 − 1) / 𝑄))
158	153, 156, 157	3eqtr2d 2786	. . . . . . . 8 ⊢ (𝜑 → (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))) = ((𝑁 − 1) / 𝑄))
159	158	breq2d 5178	. . . . . . 7 ⊢ (𝜑 → (((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ ((𝑁 − 1) / 𝑄)))
160	149, 159	bitr4d 282	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ ((𝐶↑((𝑁 − 1) / 𝑄)) − 1) ↔ ((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1)))))
161		pockthlem.11	. . . . . . 7 ⊢ (𝜑 → (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁) = 1)
162	161	breq2d 5178	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (((𝐶↑((𝑁 − 1) / 𝑄)) − 1) gcd 𝑁) ↔ 𝑃 ∥ 1))
163	147, 160, 162	3imtr3d 293	. . . . 5 ⊢ (𝜑 → (((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))) → 𝑃 ∥ 1))
164	123, 163	mtod 198	. . . 4 ⊢ (𝜑 → ¬ ((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))))
165		prmpwdvds 16951	. . . 4 ⊢ (((((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) ∈ ℤ ∧ ((od_ℤ‘𝑃)‘𝐶) ∈ ℤ) ∧ (𝑄 ∈ ℙ ∧ (𝑄 pCnt 𝐴) ∈ ℕ) ∧ (((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑(𝑄 pCnt 𝐴))) ∧ ¬ ((od_ℤ‘𝑃)‘𝐶) ∥ (((𝑁 − 1) / (𝑄↑(𝑄 pCnt 𝐴))) · (𝑄↑((𝑄 pCnt 𝐴) − 1))))) → (𝑄↑(𝑄 pCnt 𝐴)) ∥ ((od_ℤ‘𝑃)‘𝐶))
166	102, 76, 1, 4, 121, 164, 165	syl222anc 1386	. . 3 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∥ ((od_ℤ‘𝑃)‘𝐶))
167		odzphi 16843	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝐶 ∈ ℤ ∧ (𝐶 gcd 𝑃) = 1) → ((od_ℤ‘𝑃)‘𝐶) ∥ (ϕ‘𝑃))
168	10, 11, 73, 167	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∥ (ϕ‘𝑃))
169		phiprm 16824	. . . . 5 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
170	8, 169	syl 17	. . . 4 ⊢ (𝜑 → (ϕ‘𝑃) = (𝑃 − 1))
171	168, 170	breqtrd 5192	. . 3 ⊢ (𝜑 → ((od_ℤ‘𝑃)‘𝐶) ∥ (𝑃 − 1))
172	7, 76, 83, 166, 171	dvdstrd 16343	. 2 ⊢ (𝜑 → (𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑃 − 1))
173		pcdvdsb 16916	. . 3 ⊢ ((𝑄 ∈ ℙ ∧ (𝑃 − 1) ∈ ℤ ∧ (𝑄 pCnt 𝐴) ∈ ℕ₀) → ((𝑄 pCnt 𝐴) ≤ (𝑄 pCnt (𝑃 − 1)) ↔ (𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑃 − 1)))
174	1, 83, 5, 173	syl3anc 1371	. 2 ⊢ (𝜑 → ((𝑄 pCnt 𝐴) ≤ (𝑄 pCnt (𝑃 − 1)) ↔ (𝑄↑(𝑄 pCnt 𝐴)) ∥ (𝑃 − 1)))
175	172, 174	mpbird 257	1 ⊢ (𝜑 → (𝑄 pCnt 𝐴) ≤ (𝑄 pCnt (𝑃 − 1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 mod cmo 13920 ↑cexp 14112 ∥ cdvds 16302 gcd cgcd 16540 ℙcprime 16718 od_ℤcodz 16810 ϕcphi 16811 pCnt cpc 16883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-odz 16812 df-phi 16813 df-pc 16884

This theorem is referenced by: pockthg 16953

W3C validator