Theorem lighneallem2 43253

Description: Lemma 2 for lighneal 43258. (Contributed by AV, 13-Aug-2021.)

Assertion

Ref	Expression
lighneallem2	⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)

Proof of Theorem lighneallem2

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		evennn2n 15533	. . . 4 ⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁))
2	1	3ad2ant3 1128	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ 𝑁 ↔ ∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁))
3		oveq2 7024	. . . . . . . . 9 ⊢ (𝑁 = (2 · 𝑘) → (2↑𝑁) = (2↑(2 · 𝑘)))
4	3	eqcoms 2803	. . . . . . . 8 ⊢ ((2 · 𝑘) = 𝑁 → (2↑𝑁) = (2↑(2 · 𝑘)))
5		2cnd 11563	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ)
6		nncn 11494	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
7	5, 6	mulcomd 10508	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 · 𝑘) = (𝑘 · 2))
8	7	oveq2d 7032	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑(2 · 𝑘)) = (2↑(𝑘 · 2)))
9		2nn0 11762	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ0
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℕ0)
11		nnnn0 11752	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
12	5, 10, 11	expmuld 13363	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑(𝑘 · 2)) = ((2↑𝑘)↑2))
13	8, 12	eqtrd 2831	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑(2 · 𝑘)) = ((2↑𝑘)↑2))
14	13	adantl 482	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (2↑(2 · 𝑘)) = ((2↑𝑘)↑2))
15	4, 14	sylan9eqr 2853	. . . . . . 7 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → (2↑𝑁) = ((2↑𝑘)↑2))
16	15	oveq1d 7031	. . . . . 6 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → ((2↑𝑁) − 1) = (((2↑𝑘)↑2) − 1))
17	16	eqeq1d 2797	. . . . 5 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → (((2↑𝑁) − 1) = (𝑃↑𝑀) ↔ (((2↑𝑘)↑2) − 1) = (𝑃↑𝑀)))
18		elnn1uz2 12174	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 = 1 ∨ 𝑘 ∈ (ℤ≥‘2)))
19		oveq2 7024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (2↑𝑘) = (2↑1))
20		2cn 11560	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
21		exp1 13285	. . . . . . . . . . . . . . . . . 18 ⊢ (2 ∈ ℂ → (2↑1) = 2)
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (2↑1) = 2
23	19, 22	syl6eq 2847	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2↑𝑘) = 2)
24	23	oveq1d 7031	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2↑𝑘)↑2) = (2↑2))
25	24	oveq1d 7031	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (((2↑𝑘)↑2) − 1) = ((2↑2) − 1))
26		sq2 13410	. . . . . . . . . . . . . . . 16 ⊢ (2↑2) = 4
27	26	oveq1i 7026	. . . . . . . . . . . . . . 15 ⊢ ((2↑2) − 1) = (4 − 1)
28		4m1e3 11614	. . . . . . . . . . . . . . 15 ⊢ (4 − 1) = 3
29	27, 28	eqtri 2819	. . . . . . . . . . . . . 14 ⊢ ((2↑2) − 1) = 3
30	25, 29	syl6eq 2847	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (((2↑𝑘)↑2) − 1) = 3)
31	30	eqeq1d 2797	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ 3 = (𝑃↑𝑀)))
32	31	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ 3 = (𝑃↑𝑀)))
33		eqcom 2802	. . . . . . . . . . . . . 14 ⊢ (3 = (𝑃↑𝑀) ↔ (𝑃↑𝑀) = 3)
34		eldifi 4024	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
35		prmnn 15847	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
36		nnre 11493	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ)
38	37	3ad2ant1 1126	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℝ)
39		nnnn0 11752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
40	39	3ad2ant2 1127	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ0)
41	38, 40	reexpcld 13377	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ∈ ℝ)
42	41	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) ∈ ℝ)
43		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) = 3)
44	42, 43	eqled 10590	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) ≤ 3)
45	44	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃↑𝑀) = 3 → (𝑃↑𝑀) ≤ 3))
46	33, 45	syl5bi 243	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (3 = (𝑃↑𝑀) → (𝑃↑𝑀) ≤ 3))
47	35	nnred 11501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48		prmgt1 15870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
49	47, 48	jca 512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
50	34, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
51	50	3ad2ant1 1126	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
52		nnz 11853	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
53	52	3ad2ant2 1127	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℤ)
54		3rp 12245	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ+
55	54	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 3 ∈ ℝ+)
56		efexple 25539	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℝ ∧ 1 < 𝑃) ∧ 𝑀 ∈ ℤ ∧ 3 ∈ ℝ+) → ((𝑃↑𝑀) ≤ 3 ↔ 𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃)))))
57	51, 53, 55, 56	syl3anc 1364	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃↑𝑀) ≤ 3 ↔ 𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃)))))
58		oddprmge3 15873	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘3))
59		eluzle 12106	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℤ≥‘3) → 3 ≤ 𝑃)
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 3 ≤ 𝑃)
61	54	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 3 ∈ ℝ+)
62		nnrp 12250	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ+)
63	34, 35, 62	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ+)
64	61, 63	logled 24891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (3 ≤ 𝑃 ↔ (log‘3) ≤ (log‘𝑃)))
65	60, 64	mpbid 233	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘3) ≤ (log‘𝑃))
66	65	3ad2ant1 1126	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (log‘3) ≤ (log‘𝑃))
67		relogcl 24840	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ ℝ+ → (log‘3) ∈ ℝ)
68	54, 67	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (log‘3) ∈ ℝ
69		rplogcl 24868	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (log‘𝑃) ∈ ℝ+)
70	34, 49, 69	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘𝑃) ∈ ℝ+)
71	70	3ad2ant1 1126	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (log‘𝑃) ∈ ℝ+)
72		divle1le 12309	. . . . . . . . . . . . . . . . 17 ⊢ (((log‘3) ∈ ℝ ∧ (log‘𝑃) ∈ ℝ+) → (((log‘3) / (log‘𝑃)) ≤ 1 ↔ (log‘3) ≤ (log‘𝑃)))
73	68, 71, 72	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (((log‘3) / (log‘𝑃)) ≤ 1 ↔ (log‘3) ≤ (log‘𝑃)))
74	66, 73	mpbird 258	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((log‘3) / (log‘𝑃)) ≤ 1)
75		fldivle 13051	. . . . . . . . . . . . . . . . 17 ⊢ (((log‘3) ∈ ℝ ∧ (log‘𝑃) ∈ ℝ+) → (⌊‘((log‘3) / (log‘𝑃))) ≤ ((log‘3) / (log‘𝑃)))
76	68, 71, 75	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((log‘3) / (log‘𝑃))) ≤ ((log‘3) / (log‘𝑃)))
77		nnre 11493	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
78	77	3ad2ant2 1127	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
79	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘3) ∈ ℝ)
80	62	relogcld 24887	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℕ → (log‘𝑃) ∈ ℝ)
81	34, 35, 80	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘𝑃) ∈ ℝ)
82	35	nnrpd 12279	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ+)
83		1red 10488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℝ)
84	83, 48	gtned 10622	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ ℙ → 𝑃 ≠ 1)
85	82, 84	jca 512	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ+ ∧ 𝑃 ≠ 1))
86		logne0 24844	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℝ+ ∧ 𝑃 ≠ 1) → (log‘𝑃) ≠ 0)
87	34, 85, 86	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘𝑃) ≠ 0)
88	79, 81, 87	redivcld 11316	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((log‘3) / (log‘𝑃)) ∈ ℝ)
89	88	flcld 13018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℤ)
90	89	zred 11936	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℝ)
91	90	3ad2ant1 1126	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℝ)
92	88	3ad2ant1 1126	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((log‘3) / (log‘𝑃)) ∈ ℝ)
93		letr 10581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℝ ∧ (⌊‘((log‘3) / (log‘𝑃))) ∈ ℝ ∧ ((log‘3) / (log‘𝑃)) ∈ ℝ) → ((𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃))) ∧ (⌊‘((log‘3) / (log‘𝑃))) ≤ ((log‘3) / (log‘𝑃))) → 𝑀 ≤ ((log‘3) / (log‘𝑃))))
94	78, 91, 92, 93	syl3anc 1364	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃))) ∧ (⌊‘((log‘3) / (log‘𝑃))) ≤ ((log‘3) / (log‘𝑃))) → 𝑀 ≤ ((log‘3) / (log‘𝑃))))
95		1red 10488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℝ)
96		letr 10581	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℝ ∧ ((log‘3) / (log‘𝑃)) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑀 ≤ ((log‘3) / (log‘𝑃)) ∧ ((log‘3) / (log‘𝑃)) ≤ 1) → 𝑀 ≤ 1))
97	78, 92, 95, 96	syl3anc 1364	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 ≤ ((log‘3) / (log‘𝑃)) ∧ ((log‘3) / (log‘𝑃)) ≤ 1) → 𝑀 ≤ 1))
98		nnge1 11513	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
99		eqcom 2802	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 = 1 ↔ 1 = 𝑀)
100		1red 10488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℝ)
101	100, 77	letri3d 10629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℕ → (1 = 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≤ 1)))
102	99, 101	syl5rbb 285	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → ((1 ≤ 𝑀 ∧ 𝑀 ≤ 1) ↔ 𝑀 = 1))
103	102	biimpd 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → ((1 ≤ 𝑀 ∧ 𝑀 ≤ 1) → 𝑀 = 1))
104	98, 103	mpand 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → (𝑀 ≤ 1 → 𝑀 = 1))
105	104	3ad2ant2 1127	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ 1 → 𝑀 = 1))
106	97, 105	syld 47	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 ≤ ((log‘3) / (log‘𝑃)) ∧ ((log‘3) / (log‘𝑃)) ≤ 1) → 𝑀 = 1))
107	106	expd 416	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ ((log‘3) / (log‘𝑃)) → (((log‘3) / (log‘𝑃)) ≤ 1 → 𝑀 = 1)))
108	94, 107	syld 47	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃))) ∧ (⌊‘((log‘3) / (log‘𝑃))) ≤ ((log‘3) / (log‘𝑃))) → (((log‘3) / (log‘𝑃)) ≤ 1 → 𝑀 = 1)))
109	76, 108	mpan2d 690	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃))) → (((log‘3) / (log‘𝑃)) ≤ 1 → 𝑀 = 1)))
110	74, 109	mpid 44	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃))) → 𝑀 = 1))
111	57, 110	sylbid 241	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃↑𝑀) ≤ 3 → 𝑀 = 1))
112	46, 111	syld 47	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (3 = (𝑃↑𝑀) → 𝑀 = 1))
113	112	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (3 = (𝑃↑𝑀) → 𝑀 = 1))
114	32, 113	sylbid 241	. . . . . . . . . 10 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
115	114	ex 413	. . . . . . . . 9 ⊢ (𝑘 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
116		sq1 13408	. . . . . . . . . . . . . 14 ⊢ (1↑2) = 1
117	116	eqcomi 2804	. . . . . . . . . . . . 13 ⊢ 1 = (1↑2)
118	117	oveq2i 7027	. . . . . . . . . . . 12 ⊢ (((2↑𝑘)↑2) − 1) = (((2↑𝑘)↑2) − (1↑2))
119	118	eqeq1i 2800	. . . . . . . . . . 11 ⊢ ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ (((2↑𝑘)↑2) − (1↑2)) = (𝑃↑𝑀))
120		eqcom 2802	. . . . . . . . . . . 12 ⊢ ((((2↑𝑘)↑2) − (1↑2)) = (𝑃↑𝑀) ↔ (𝑃↑𝑀) = (((2↑𝑘)↑2) − (1↑2)))
121	9	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘2) → 2 ∈ ℕ0)
122		eluzge2nn0 12136	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℕ0)
123	121, 122	nn0expcld 13457	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘2) → (2↑𝑘) ∈ ℕ0)
124	123	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (2↑𝑘) ∈ ℕ0)
125		1nn0 11761	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ0
126	125	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → 1 ∈ ℕ0)
127		1p1e2 11610	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
128	22	eqcomi 2804	. . . . . . . . . . . . . . . . 17 ⊢ 2 = (2↑1)
129	127, 128	eqtri 2819	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = (2↑1)
130		eluz2gt1 12169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 𝑘)
131		2re 11559	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
132	131	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ≥‘2) → 2 ∈ ℝ)
133		1zzd 11862	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 ∈ ℤ)
134		eluzelz 12103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ≥‘2) → 𝑘 ∈ ℤ)
135		1lt2 11656	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
136	135	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ≥‘2) → 1 < 2)
137	132, 133, 134, 136	ltexp2d 13464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘2) → (1 < 𝑘 ↔ (2↑1) < (2↑𝑘)))
138	130, 137	mpbid 233	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ≥‘2) → (2↑1) < (2↑𝑘))
139	129, 138	eqbrtrid 4997	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘2) → (1 + 1) < (2↑𝑘))
140	139	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (1 + 1) < (2↑𝑘))
141	34, 39	anim12i 612	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ0))
142	141	3adant3 1125	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ0))
143	142	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ0))
144		difsqpwdvds 16052	. . . . . . . . . . . . . 14 ⊢ ((((2↑𝑘) ∈ ℕ0 ∧ 1 ∈ ℕ0 ∧ (1 + 1) < (2↑𝑘)) ∧ (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ0)) → ((𝑃↑𝑀) = (((2↑𝑘)↑2) − (1↑2)) → 𝑃 ∥ (2 · 1)))
145	124, 126, 140, 143, 144	syl31anc 1366	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (((2↑𝑘)↑2) − (1↑2)) → 𝑃 ∥ (2 · 1)))
146		2t1e2 11648	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 1) = 2
147	146	breq2i 4970	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∥ (2 · 1) ↔ 𝑃 ∥ 2)
148		prmuz2 15869	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
149	34, 148	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ≥‘2))
150		2prm 15865	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℙ
151		dvdsprm 15876	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
152	149, 150, 151	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
153	147, 152	syl5bb 284	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 · 1) ↔ 𝑃 = 2))
154		eldifsn 4626	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
155		eqneqall 2995	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 ≠ 2 → 𝑀 = 1))
156	155	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ≠ 2 → (𝑃 = 2 → 𝑀 = 1))
157	154, 156	simplbiim 505	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 = 2 → 𝑀 = 1))
158	153, 157	sylbid 241	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 · 1) → 𝑀 = 1))
159	158	3ad2ant1 1126	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (2 · 1) → 𝑀 = 1))
160	159	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑃 ∥ (2 · 1) → 𝑀 = 1))
161	145, 160	syld 47	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (((2↑𝑘)↑2) − (1↑2)) → 𝑀 = 1))
162	120, 161	syl5bi 243	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − (1↑2)) = (𝑃↑𝑀) → 𝑀 = 1))
163	119, 162	syl5bi 243	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
164	163	ex 413	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘2) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
165	115, 164	jaoi 852	. . . . . . . 8 ⊢ ((𝑘 = 1 ∨ 𝑘 ∈ (ℤ≥‘2)) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
166	18, 165	sylbi 218	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
167	166	impcom 408	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
168	167	adantr 481	. . . . 5 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
169	17, 168	sylbid 241	. . . 4 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
170	169	rexlimdva2 3250	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
171	2, 170	sylbid 241	. 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ 𝑁 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
172	171	3imp 1104	1 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∃wrex 3106   ∖ cdif 3856  {csn 4472   class class class wbr 4962  ‘cfv 6225  (class class class)co 7016  ℂcc 10381  ℝcr 10382  0cc0 10383  1c1 10384   + caddc 10386   · cmul 10388   < clt 10521   ≤ cle 10522   − cmin 10717   / cdiv 11145  ℕcn 11486  2c2 11540  3c3 11541  4c4 11542  ℕ₀cn0 11745  ℤcz 11829  ℤ_≥cuz 12093  ℝ⁺crp 12239  ⌊cfl 13010  ↑cexp 13279   ∥ cdvds 15440  ℙcprime 15844  logclog 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462  ax-mulf 10463

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-fi 8721  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-ioc 12593  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-mod 13088  df-seq 13220  df-exp 13280  df-fac 13484  df-bc 13513  df-hash 13541  df-shft 14260  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-limsup 14662  df-clim 14679  df-rlim 14680  df-sum 14877  df-ef 15254  df-sin 15256  df-cos 15257  df-pi 15259  df-dvds 15441  df-gcd 15677  df-prm 15845  df-pc 16003  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-starv 16409  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-hom 16418  df-cco 16419  df-rest 16525  df-topn 16526  df-0g 16544  df-gsum 16545  df-topgen 16546  df-pt 16547  df-prds 16550  df-xrs 16604  df-qtop 16609  df-imas 16610  df-xps 16612  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-mulg 17982  df-cntz 18188  df-cmn 18635  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-fbas 20224  df-fg 20225  df-cnfld 20228  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-cld 21311  df-ntr 21312  df-cls 21313  df-nei 21390  df-lp 21428  df-perf 21429  df-cn 21519  df-cnp 21520  df-haus 21607  df-tx 21854  df-hmeo 22047  df-fil 22138  df-fm 22230  df-flim 22231  df-flf 22232  df-xms 22613  df-ms 22614  df-tms 22615  df-cncf 23169  df-limc 24147  df-dv 24148  df-log 24821

This theorem is referenced by:  lighneal  43258