lighneallem2 - Mathbox for Alexander van der Vekens

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Theorem lighneallem2 47530

Description: Lemma 2 for lighneal 47535. (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 16384	. . . 4 ⊢ (𝑁 ∈ ℕ → (2 ∥ 𝑁 ↔ ∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁))
2	1	3ad2ant3 1134	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ 𝑁 ↔ ∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁))
3		oveq2 7438	. . . . . . . . 9 ⊢ (𝑁 = (2 · 𝑘) → (2↑𝑁) = (2↑(2 · 𝑘)))
4	3	eqcoms 2742	. . . . . . . 8 ⊢ ((2 · 𝑘) = 𝑁 → (2↑𝑁) = (2↑(2 · 𝑘)))
5		2cnd 12341	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℂ)
6		nncn 12271	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
7	5, 6	mulcomd 11279	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → (2 · 𝑘) = (𝑘 · 2))
8	7	oveq2d 7446	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑(2 · 𝑘)) = (2↑(𝑘 · 2)))
9		2nn0 12540	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
10	9	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 2 ∈ ℕ₀)
11		nnnn0 12530	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
12	5, 10, 11	expmuld 14185	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → (2↑(𝑘 · 2)) = ((2↑𝑘)↑2))
13	8, 12	eqtrd 2774	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (2↑(2 · 𝑘)) = ((2↑𝑘)↑2))
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → (2↑(2 · 𝑘)) = ((2↑𝑘)↑2))
15	4, 14	sylan9eqr 2796	. . . . . . 7 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → (2↑𝑁) = ((2↑𝑘)↑2))
16	15	oveq1d 7445	. . . . . 6 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → ((2↑𝑁) − 1) = (((2↑𝑘)↑2) − 1))
17	16	eqeq1d 2736	. . . . 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 12964	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ ↔ (𝑘 = 1 ∨ 𝑘 ∈ (ℤ_≥‘2)))
19		oveq2 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 1 → (2↑𝑘) = (2↑1))
20		2cn 12338	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
21		exp1 14104	. . . . . . . . . . . . . . . . . 18 ⊢ (2 ∈ ℂ → (2↑1) = 2)
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (2↑1) = 2
23	19, 22	eqtrdi 2790	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 1 → (2↑𝑘) = 2)
24	23	oveq1d 7445	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 1 → ((2↑𝑘)↑2) = (2↑2))
25	24	oveq1d 7445	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 1 → (((2↑𝑘)↑2) − 1) = ((2↑2) − 1))
26		sq2 14232	. . . . . . . . . . . . . . . 16 ⊢ (2↑2) = 4
27	26	oveq1i 7440	. . . . . . . . . . . . . . 15 ⊢ ((2↑2) − 1) = (4 − 1)
28		4m1e3 12392	. . . . . . . . . . . . . . 15 ⊢ (4 − 1) = 3
29	27, 28	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ ((2↑2) − 1) = 3
30	25, 29	eqtrdi 2790	. . . . . . . . . . . . 13 ⊢ (𝑘 = 1 → (((2↑𝑘)↑2) − 1) = 3)
31	30	eqeq1d 2736	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ 3 = (𝑃↑𝑀)))
32	31	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ 3 = (𝑃↑𝑀)))
33		eqcom 2741	. . . . . . . . . . . . . 14 ⊢ (3 = (𝑃↑𝑀) ↔ (𝑃↑𝑀) = 3)
34		eldifi 4140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
35		prmnn 16707	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
36		nnre 12270	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ)
38	37	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑃 ∈ ℝ)
39		nnnn0 12530	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40	39	3ad2ant2 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
41	38, 40	reexpcld 14199	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃↑𝑀) ∈ ℝ)
42	41	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) ∈ ℝ)
43		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) = 3)
44	42, 43	eqled 11361	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝑃↑𝑀) = 3) → (𝑃↑𝑀) ≤ 3)
45	44	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃↑𝑀) = 3 → (𝑃↑𝑀) ≤ 3))
46	33, 45	biimtrid 242	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (3 = (𝑃↑𝑀) → (𝑃↑𝑀) ≤ 3))
47	35	nnred 12278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ)
48		prmgt1 16730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ ℙ → 1 < 𝑃)
49	47, 48	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
50	34, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
51	50	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ ℝ ∧ 1 < 𝑃))
52		nnz 12631	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
53	52	3ad2ant2 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℤ)
54		3rp 13037	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ⁺
55	54	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 3 ∈ ℝ⁺)
56		efexple 27339	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℝ ∧ 1 < 𝑃) ∧ 𝑀 ∈ ℤ ∧ 3 ∈ ℝ⁺) → ((𝑃↑𝑀) ≤ 3 ↔ 𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃)))))
57	51, 53, 55, 56	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑃↑𝑀) ≤ 3 ↔ 𝑀 ≤ (⌊‘((log‘3) / (log‘𝑃)))))
58		oddprmge3 16733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘3))
59		eluzle 12888	. . . . . . . . . . . . . . . . . . 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 13043	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ ℕ → 𝑃 ∈ ℝ⁺)
63	34, 35, 62	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ⁺)
64	61, 63	logled 26683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (3 ≤ 𝑃 ↔ (log‘3) ≤ (log‘𝑃)))
65	60, 64	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘3) ≤ (log‘𝑃))
66	65	3ad2ant1 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (log‘3) ≤ (log‘𝑃))
67		relogcl 26631	. . . . . . . . . . . . . . . . . 18 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
68	54, 67	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (log‘3) ∈ ℝ
69		rplogcl 26660	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ ℝ ∧ 1 < 𝑃) → (log‘𝑃) ∈ ℝ⁺)
70	34, 49, 69	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘𝑃) ∈ ℝ⁺)
71	70	3ad2ant1 1132	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (log‘𝑃) ∈ ℝ⁺)
72		divle1le 13102	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((log‘3) / (log‘𝑃)) ≤ 1)
75		fldivle 13867	. . . . . . . . . . . . . . . . 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 12270	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
78	77	3ad2ant2 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
79	68	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘3) ∈ ℝ)
80	62	relogcld 26679	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℕ → (log‘𝑃) ∈ ℝ)
81	34, 35, 80	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (log‘𝑃) ∈ ℝ)
82	35	nnrpd 13072	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ⁺)
83		1red 11259	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ ℙ → 1 ∈ ℝ)
84	83, 48	gtned 11393	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ ℙ → 𝑃 ≠ 1)
85	82, 84	jca 511	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 ∈ ℙ → (𝑃 ∈ ℝ⁺ ∧ 𝑃 ≠ 1))
86		logne0 26635	. . . . . . . . . . . . . . . . . . . . . . 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 12092	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((log‘3) / (log‘𝑃)) ∈ ℝ)
89	88	flcld 13834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℤ)
90	89	zred 12719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℝ)
91	90	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (⌊‘((log‘3) / (log‘𝑃))) ∈ ℝ)
92	88	3ad2ant1 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((log‘3) / (log‘𝑃)) ∈ ℝ)
93		letr 11352	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 11259	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 1 ∈ ℝ)
96		letr 11352	. . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑀 ≤ ((log‘3) / (log‘𝑃)) ∧ ((log‘3) / (log‘𝑃)) ≤ 1) → 𝑀 ≤ 1))
98		nnge1 12291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
99		eqcom 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 = 1 ↔ 1 = 𝑀)
100		1red 11259	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℝ)
101	100, 77	letri3d 11400	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℕ → (1 = 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≤ 1)))
102	99, 101	bitr2id 284	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → ((1 ≤ 𝑀 ∧ 𝑀 ≤ 1) ↔ 𝑀 = 1))
103	102	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ → ((1 ≤ 𝑀 ∧ 𝑀 ≤ 1) → 𝑀 = 1))
104	98, 103	mpand 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ → (𝑀 ≤ 1 → 𝑀 = 1))
105	104	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (3 = (𝑃↑𝑀) → 𝑀 = 1))
114	32, 113	sylbid 240	. . . . . . . . . 10 ⊢ ((𝑘 = 1 ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
115	114	ex 412	. . . . . . . . 9 ⊢ (𝑘 = 1 → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
116		sq1 14230	. . . . . . . . . . . . . 14 ⊢ (1↑2) = 1
117	116	eqcomi 2743	. . . . . . . . . . . . 13 ⊢ 1 = (1↑2)
118	117	oveq2i 7441	. . . . . . . . . . . 12 ⊢ (((2↑𝑘)↑2) − 1) = (((2↑𝑘)↑2) − (1↑2))
119	118	eqeq1i 2739	. . . . . . . . . . 11 ⊢ ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) ↔ (((2↑𝑘)↑2) − (1↑2)) = (𝑃↑𝑀))
120		eqcom 2741	. . . . . . . . . . . 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 12926	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ₀)
123	121, 122	nn0expcld 14281	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (2↑𝑘) ∈ ℕ₀)
124	123	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (2↑𝑘) ∈ ℕ₀)
125		1nn0 12539	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℕ₀
126	125	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → 1 ∈ ℕ₀)
127		1p1e2 12388	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
128	22	eqcomi 2743	. . . . . . . . . . . . . . . . 17 ⊢ 2 = (2↑1)
129	127, 128	eqtri 2762	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = (2↑1)
130		eluz2gt1 12959	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 1 < 𝑘)
131		2re 12337	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ
132	131	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 2 ∈ ℝ)
133		1zzd 12645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 1 ∈ ℤ)
134		eluzelz 12885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℤ)
135		1lt2 12434	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 2
136	135	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 1 < 2)
137	132, 133, 134, 136	ltexp2d 14286	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (1 < 𝑘 ↔ (2↑1) < (2↑𝑘)))
138	130, 137	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (2↑1) < (2↑𝑘))
139	129, 138	eqbrtrid 5182	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (1 + 1) < (2↑𝑘))
140	139	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (1 + 1) < (2↑𝑘))
141	34, 39	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ₀))
142	141	3adant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ₀))
143	142	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → (𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ₀))
144		difsqpwdvds 16920	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((𝑃↑𝑀) = (((2↑𝑘)↑2) − (1↑2)) → 𝑃 ∥ (2 · 1)))
146		2t1e2 12426	. . . . . . . . . . . . . . . . . 18 ⊢ (2 · 1) = 2
147	146	breq2i 5155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∥ (2 · 1) ↔ 𝑃 ∥ 2)
148		prmuz2 16729	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
149	34, 148	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
150		2prm 16725	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℙ
151		dvdsprm 16736	. . . . . . . . . . . . . . . . . 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	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 · 1) ↔ 𝑃 = 2))
154		eldifsn 4790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
155		eqneqall 2948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 = 2 → (𝑃 ≠ 2 → 𝑀 = 1))
156	155	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ≠ 2 → (𝑃 = 2 → 𝑀 = 1))
157	154, 156	simplbiim 504	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 = 2 → 𝑀 = 1))
158	153, 157	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 · 1) → 𝑀 = 1))
159	158	3ad2ant1 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (2 · 1) → 𝑀 = 1))
160	159	adantl 481	. . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − (1↑2)) = (𝑃↑𝑀) → 𝑀 = 1))
163	119, 162	biimtrid 242	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ_≥‘2) ∧ (𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ)) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
164	163	ex 412	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘2) → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
165	115, 164	jaoi 857	. . . . . . . 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 217	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
167	166	impcom 407	. . . . . 6 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) → ((((2↑𝑘)↑2) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
168	167	adantr 480	. . . . 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 240	. . . 4 ⊢ ((((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 𝑘 ∈ ℕ) ∧ (2 · 𝑘) = 𝑁) → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1))
170	169	rexlimdva2 3154	. . 3 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (∃𝑘 ∈ ℕ (2 · 𝑘) = 𝑁 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
171	2, 170	sylbid 240	. 2 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2 ∥ 𝑁 → (((2↑𝑁) − 1) = (𝑃↑𝑀) → 𝑀 = 1)))
172	171	3imp 1110	1 ⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∖ cdif 3959 {csn 4630 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 ℕcn 12263 2c2 12318 3c3 12319 4c4 12320 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ℝ⁺crp 13031 ⌊cfl 13826 ↑cexp 14098 ∥ cdvds 16286 ℙcprime 16704 logclog 26610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-dvds 16287 df-gcd 16528 df-prm 16705 df-pc 16870 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612

This theorem is referenced by: lighneal 47535

W3C validator