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Theorem rplogsumlem2 27412

Description: Lemma for rplogsum 27454. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)

**Assertion**
Ref	Expression
rplogsumlem2	⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)

Distinct variable group: 𝐴,𝑛

Proof of Theorem rplogsumlem2 Dummy variables 𝑘 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flid 13730	. . . . 5 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2	1	oveq2d 7369	. . . 4 ⊢ (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
3	2	sumeq1d 15625	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛))
4		fveq2 6826	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → (Λ‘𝑛) = (Λ‘(𝑝↑𝑘)))
5		eleq1 2816	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (𝑛 ∈ ℙ ↔ (𝑝↑𝑘) ∈ ℙ))
6		fveq2 6826	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (log‘𝑛) = (log‘(𝑝↑𝑘)))
7	5, 6	ifbieq1d 4503	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0))
8	4, 7	oveq12d 7371	. . . . 5 ⊢ (𝑛 = (𝑝↑𝑘) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = ((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)))
9		id 22	. . . . 5 ⊢ (𝑛 = (𝑝↑𝑘) → 𝑛 = (𝑝↑𝑘))
10	8, 9	oveq12d 7371	. . . 4 ⊢ (𝑛 = (𝑝↑𝑘) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)))
11		zre 12493	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12		elfznn 13474	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14		vmacl 27044	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
16	13	nnrpd 12953	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
17	16	relogcld 26548	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18		0re 11136	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4524	. . . . . . . 8 ⊢ (((log‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
20	17, 18, 19	sylancl 586	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
21	15, 20	resubcld 11566	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
22	21, 13	nndivred 12200	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
23	22	recnd 11162	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℂ)
24		simprr 772	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
25		vmaprm 27043	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26		prmnn 16603	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	nnred 12161	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28		prmgt1 16626	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 1 < 𝑛)
29	27, 28	rplogcld 26554	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ⁺)
30	25, 29	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ⁺)
31	30	rpne0d 12960	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
32	31	necon2bi 2955	. . . . . . . . . 10 ⊢ ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
33	32	ad2antll 729	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4489	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
35	24, 34	oveq12d 7371	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36		0m0e0 12261	. . . . . . 7 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2780	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
38	37	oveq1d 7368	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (0 / 𝑛))
39	12	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
40	39	nnrpd 12953	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ⁺)
41	40	rpcnne0d 12964	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42		div0 11830	. . . . . 6 ⊢ ((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) → (0 / 𝑛) = 0)
43	41, 42	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
44	38, 43	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = 0)
45	10, 11, 23, 44	fsumvma2 27141	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...(⌊‘𝐴))(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)))
46	3, 45	eqtr3d 2766	. 2 ⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)))
47		fzfid 13898	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2...((abs‘𝐴) + 1)) ∈ Fin)
48		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
49	48	elin2d 4158	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
50		prmnn 16603	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
52	51	nnred 12161	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
53	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
54		zcn 12494	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
55	54	abscld 15364	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
56		peano2re 11307	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
57	55, 56	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℝ)
58	57	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℝ)
59		elinel1 4154	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
60		elicc2 13332	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
61	18, 11, 60	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
62	59, 61	imbitrid 244	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
63	62	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
64	63	simp3d 1144	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
65	54	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℂ)
66	65	abscld 15364	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
67	53	leabsd 15340	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
68	66	lep1d 12074	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
69	53, 66, 58, 67, 68	letrd 11291	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
70	52, 53, 58, 64, 69	letrd 11291	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
71		prmuz2 16625	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
72	49, 71	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
73		nn0abscl 15237	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ₀)
74		nn0p1nn 12441	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) ∈ ℕ₀ → ((abs‘𝐴) + 1) ∈ ℕ)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
76	75	nnzd 12516	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
77	76	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
78		elfz5 13437	. . . . . . . . 9 ⊢ ((𝑝 ∈ (ℤ_≥‘2) ∧ ((abs‘𝐴) + 1) ∈ ℤ) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
79	72, 77, 78	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
80	70, 79	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (2...((abs‘𝐴) + 1)))
81	80	ex 412	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (2...((abs‘𝐴) + 1))))
82	81	ssrdv 3943	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
83	47, 82	ssfid 9170	. . . 4 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
84		fzfid 13898	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
85		simprl 770	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
86	85	elin2d 4158	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
87		elfznn 13474	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
88	87	ad2antll 729	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
89		vmappw 27042	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
90	86, 88, 89	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
91	51	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
92	91	nnrpd 12953	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ⁺)
93	92	relogcld 26548	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
94	90, 93	eqeltrd 2828	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) ∈ ℝ)
95	88	nnnn0d 12463	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ₀)
96		nnexpcl 13999	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑝↑𝑘) ∈ ℕ)
97	91, 95, 96	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℕ)
98	97	nnrpd 12953	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℝ⁺)
99	98	relogcld 26548	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝↑𝑘)) ∈ ℝ)
100		ifcl 4524	. . . . . . . . 9 ⊢ (((log‘(𝑝↑𝑘)) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) ∈ ℝ)
101	99, 18, 100	sylancl 586	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) ∈ ℝ)
102	94, 101	resubcld 11566	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → ((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) ∈ ℝ)
103	102, 97	nndivred 12200	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℝ)
104	103	anassrs 467	. . . . 5 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℝ)
105	84, 104	fsumrecl 15659	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℝ)
106	83, 105	fsumrecl 15659	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℝ)
107	51	nnrpd 12953	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
108	107	relogcld 26548	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
109		uz2m1nn 12842	. . . . . . 7 ⊢ (𝑝 ∈ (ℤ_≥‘2) → (𝑝 − 1) ∈ ℕ)
110	72, 109	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℕ)
111	51, 110	nnmulcld 12199	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
112	108, 111	nndivred 12200	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
113	83, 112	fsumrecl 15659	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
114		2re 12220	. . . 4 ⊢ 2 ∈ ℝ
115	114	a1i 11	. . 3 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℝ)
116	18	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
117	51	nngt0d 12195	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
118	116, 52, 53, 117, 64	ltletrd 11294	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
119	53, 118	elrpd 12952	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
120	119	relogcld 26548	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
121		prmgt1 16626	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
122	49, 121	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
123	52, 122	rplogcld 26554	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
124	120, 123	rerpdivcld 12986	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
125	123	rpcnd 12957	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
126	125	mullidd 11152	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
127	107, 119	logled 26552	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ≤ 𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
128	64, 127	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
129	126, 128	eqbrtrd 5117	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
130		1re 11134	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
131	130	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
132	131, 120, 123	lemuldivd 13004	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
133	129, 132	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
134		flge1nn 13743	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
135	124, 133, 134	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
136		nnuz 12796	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
137	135, 136	eleqtrdi 2838	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1))
138	103	recnd 11162	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℂ)
139	138	anassrs 467	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ∈ ℂ)
140		oveq2 7361	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑝↑𝑘) = (𝑝↑1))
141	140	fveq2d 6830	. . . . . . . . 9 ⊢ (𝑘 = 1 → (Λ‘(𝑝↑𝑘)) = (Λ‘(𝑝↑1)))
142	140	eleq1d 2813	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑝↑𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
143	140	fveq2d 6830	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (log‘(𝑝↑𝑘)) = (log‘(𝑝↑1)))
144	142, 143	ifbieq1d 4503	. . . . . . . . 9 ⊢ (𝑘 = 1 → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
145	141, 144	oveq12d 7371	. . . . . . . 8 ⊢ (𝑘 = 1 → ((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) = ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)))
146	145, 140	oveq12d 7371	. . . . . . 7 ⊢ (𝑘 = 1 → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) = (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)))
147	137, 139, 146	fsum1p 15678	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) = ((((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) + Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘))))
148	51	nncnd 12162	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
149	148	exp1d 14066	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
150	149	fveq2d 6830	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
151		vmaprm 27043	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → (Λ‘𝑝) = (log‘𝑝))
152	49, 151	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑝) = (log‘𝑝))
153	150, 152	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
154	149, 49	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
155	154	iftrued 4486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
156	149	fveq2d 6830	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
157	155, 156	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
158	153, 157	oveq12d 7371	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = ((log‘𝑝) − (log‘𝑝)))
159	125	subidd 11481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
160	158, 159	eqtrd 2764	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
161	160, 149	oveq12d 7371	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = (0 / 𝑝))
162	107	rpcnne0d 12964	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
163		div0 11830	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (0 / 𝑝) = 0)
164	162, 163	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 / 𝑝) = 0)
165	161, 164	eqtrd 2764	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
166		1p1e2 12266	. . . . . . . . . 10 ⊢ (1 + 1) = 2
167	166	oveq1i 7363	. . . . . . . . 9 ⊢ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝))))
168	167	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝)))))
169		elfzuz 13441	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
170		eluz2nn 12807	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
171	169, 170	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
172	171, 167	eleq2s 2846	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
173	49, 172, 89	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
174	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℕ)
175		nnq 12881	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ → 𝑝 ∈ ℚ)
176	174, 175	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℚ)
177	169, 167	eleq2s 2846	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
178	177	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑘 ∈ (ℤ_≥‘2))
179		expnprm 16832	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℚ ∧ 𝑘 ∈ (ℤ_≥‘2)) → ¬ (𝑝↑𝑘) ∈ ℙ)
180	176, 178, 179	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ¬ (𝑝↑𝑘) ∈ ℙ)
181	180	iffalsed 4489	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = 0)
182	173, 181	oveq12d 7371	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) = ((log‘𝑝) − 0))
183	125	subid1d 11482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − 0) = (log‘𝑝))
184	183	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) − 0) = (log‘𝑝))
185	182, 184	eqtrd 2764	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) = (log‘𝑝))
186	185	oveq1d 7368	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) = ((log‘𝑝) / (𝑝↑𝑘)))
187	168, 186	sumeq12dv 15631	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)))
188	165, 187	oveq12d 7371	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) + Σ𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘))) = (0 + Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘))))
189		fzfid 13898	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
190	108	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ)
191		nnnn0 12409	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
192	51, 191, 96	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℕ)
193	190, 192	nndivred 12200	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
194	171, 193	sylan2 593	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
195	189, 194	fsumrecl 15659	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
196	195	recnd 11162	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℂ)
197	196	addlidd 11335	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 + Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘))) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)))
198	147, 188, 197	3eqtrd 2768	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)))
199	107	rpreccld 12965	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ⁺)
200	124	flcld 13720	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
201	200	peano2zd 12601	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
202	199, 201	rpexpcld 14172	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ⁺)
203	202	rpge0d 12959	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
204	51	nnrecred 12197	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
205	204	resqcld 14050	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
206	135	peano2nnd 12163	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
207	206	nnnn0d 12463	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ₀)
208	204, 207	reexpcld 14088	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
209	205, 208	subge02d 11730	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2)))
210	203, 209	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2))
211	110	nnrpd 12953	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ⁺)
212	211	rpcnne0d 12964	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
213	199	rpcnd 12957	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
214		dmdcan 11852	. . . . . . . . . . 11 ⊢ ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ (1 / 𝑝) ∈ ℂ) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
215	212, 162, 213, 214	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
216	131	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
217		divsubdir 11836	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
218	148, 216, 162, 217	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
219		divid 11828	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
220	162, 219	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 / 𝑝) = 1)
221	220	oveq1d 7368	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
222	218, 221	eqtrd 2764	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
223		divdiv1 11853	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
224	216, 162, 212, 223	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
225	222, 224	oveq12d 7371	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
226	51	nnne0d 12196	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
227	213, 148, 226	divrecd 11921	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
228	213	sqvald 14068	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
229	227, 228	eqtr4d 2767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
230	215, 225, 229	3eqtr3d 2772	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
231	210, 230	breqtrrd 5123	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
232	205, 208	resubcld 11566	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
233	111	nnrecred 12197	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
234		resubcl 11446	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑝) ∈ ℝ) → (1 − (1 / 𝑝)) ∈ ℝ)
235	130, 204, 234	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ)
236		recgt1 12039	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
237	52, 117, 236	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
238	122, 237	mpbid 232	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) < 1)
239		posdif 11631	. . . . . . . . . . 11 ⊢ (((1 / 𝑝) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
240	204, 130, 239	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
241	238, 240	mpbid 232	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < (1 − (1 / 𝑝)))
242		ledivmul 12019	. . . . . . . . 9 ⊢ (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ ∧ (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ ∧ ((1 − (1 / 𝑝)) ∈ ℝ ∧ 0 < (1 − (1 / 𝑝)))) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1))))))
243	232, 233, 235, 241, 242	syl112anc 1376	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1))))))
244	231, 243	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))))
245	235, 241	elrpd 12952	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ⁺)
246	232, 245	rerpdivcld 12986	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ∈ ℝ)
247	246, 233, 123	lemul2d 12999	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))) ↔ ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))) ≤ ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1))))))
248	244, 247	mpbid 232	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))) ≤ ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
249	125	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℂ)
250	192	nncnd 12162	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℂ)
251	192	nnne0d 12196	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ≠ 0)
252	249, 250, 251	divrecd 11921	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
253	148	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ)
254	51	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ)
255	254	nnne0d 12196	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
256		nnz 12510	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
257	256	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
258	253, 255, 257	exprecd 14079	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝↑𝑘)))
259	258	oveq2d 7369	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
260	252, 259	eqtr4d 2767	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
261	171, 260	sylan2 593	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
262	261	sumeq2dv 15627	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
263	171	nnnn0d 12463	. . . . . . . . 9 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ₀)
264		expcl 14004	. . . . . . . . 9 ⊢ (((1 / 𝑝) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
265	213, 263, 264	syl2an 596	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
266	189, 125, 265	fsummulc2 15709	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
267		fzval3 13655	. . . . . . . . . . 11 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
268	200, 267	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
269	268	sumeq1d 15625	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = Σ𝑘 ∈ (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘))
270	204, 238	ltned 11270	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
271		2nn0 12419	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
272	271	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ₀)
273		eluzp1p1 12781	. . . . . . . . . . . 12 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
274	137, 273	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
275		df-2 12209	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
276	275	fveq2i 6829	. . . . . . . . . . 11 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
277	274, 276	eleqtrrdi 2839	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘2))
278	213, 270, 272, 277	geoserg 15791	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))((1 / 𝑝)↑𝑘) = ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))))
279	269, 278	eqtrd 2764	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘) = ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))))
280	279	oveq2d 7369	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) · Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))))
281	262, 266, 280	3eqtr2d 2770	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝)))))
282	111	nncnd 12162	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
283	111	nnne0d 12196	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
284	125, 282, 283	divrecd 11921	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
285	248, 281, 284	3brtr4d 5127	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
286	198, 285	eqbrtrd 5117	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
287	83, 105, 112, 286	fsumle 15724	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ≤ Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))))
288		elfzuz 13441	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ_≥‘2))
289		eluz2nn 12807	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℕ)
290	288, 289	syl 17	. . . . . . . . . 10 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ ℕ)
291	290	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℕ)
292	291	nnred 12161	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℝ)
293	288	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ (ℤ_≥‘2))
294		eluz2gt1 12839	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
295	293, 294	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 1 < 𝑝)
296	292, 295	rplogcld 26554	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (log‘𝑝) ∈ ℝ⁺)
297	293, 109	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 − 1) ∈ ℕ)
298	291, 297	nnmulcld 12199	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
299	298	nnrpd 12953	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ⁺)
300	296, 299	rpdivcld 12972	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ⁺)
301	300	rpred 12955	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
302	47, 301	fsumrecl 15659	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
303	300	rpge0d 12959	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
304	47, 301, 303, 82	fsumless 15721	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
305		rplogsumlem1 27411	. . . . 5 ⊢ (((abs‘𝐴) + 1) ∈ ℕ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
306	75, 305	syl 17	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
307	113, 302, 115, 304, 306	letrd 11291	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
308	106, 113, 115, 287, 307	letrd 11291	. 2 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(((Λ‘(𝑝↑𝑘)) − if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0)) / (𝑝↑𝑘)) ≤ 2)
309	46, 308	eqbrtrd 5117	1 ⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∩ cin 3904 ifcif 4478 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 ℕcn 12146 2c2 12201 ℕ₀cn0 12402 ℤcz 12489 ℤ_≥cuz 12753 ℚcq 12867 ℝ⁺crp 12911 [,]cicc 13269 ...cfz 13428 ..^cfzo 13575 ⌊cfl 13712 ↑cexp 13986 abscabs 15159 Σcsu 15611 ℙcprime 16600 logclog 26479 Λcvma 27018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-dvds 16182 df-gcd 16424 df-prm 16601 df-pc 16767 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-cxp 26482 df-vma 27024

This theorem is referenced by: rplogsum 27454

W3C validator