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Theorem rplogsumlem2 27686

Description: Lemma for rplogsum 27728. 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 13861	. . . . 5 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2	1	oveq2d 7439	. . . 4 ⊢ (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
3	2	sumeq1d 15777	. . 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 6888	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → (Λ‘𝑛) = (Λ‘(𝑝↑𝑘)))
5		eleq1 2854	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (𝑛 ∈ ℙ ↔ (𝑝↑𝑘) ∈ ℙ))
6		fveq2 6888	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (log‘𝑛) = (log‘(𝑝↑𝑘)))
7	5, 6	ifbieq1d 4517	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0))
8	4, 7	oveq12d 7441	. . . . 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 23	. . . . 5 ⊢ (𝑛 = (𝑝↑𝑘) → 𝑛 = (𝑝↑𝑘))
10	8, 9	oveq12d 7441	. . . 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 12613	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12		elfznn 13600	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
13	12	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14		vmacl 27319	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
16	13	nnrpd 13076	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
17	16	relogcld 26825	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18		0re 11228	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4538	. . . . . . . 8 ⊢ (((log‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
20	17, 18, 19	sylancl 598	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
21	15, 20	resubcld 11660	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
22	21, 13	nndivred 12308	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
23	22	recnd 11255	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℂ)
24		simprr 785	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
25		vmaprm 27318	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26		prmnn 16757	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	nnred 12266	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28		prmgt1 16781	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 1 < 𝑛)
29	27, 28	rplogcld 26831	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ⁺)
30	25, 29	eqeltrd 2866	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ⁺)
31	30	rpne0d 13083	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
32	31	necon2bi 2991	. . . . . . . . . 10 ⊢ ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
33	32	ad2antll 742	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4503	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
35	24, 34	oveq12d 7441	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36		0m0e0 12377	. . . . . . 7 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2817	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
38	37	oveq1d 7438	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (0 / 𝑛))
39	12	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
40	39	nnrpd 13076	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ⁺)
41	40	rpcnne0d 13087	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42		div0 11921	. . . . . 6 ⊢ ((𝑛 ∈ ℂ ∧ 𝑛 ≠ 0) → (0 / 𝑛) = 0)
43	41, 42	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (0 / 𝑛) = 0)
44	38, 43	eqtrd 2801	. . . 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 27415	. . 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 2803	. 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 14029	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2...((abs‘𝐴) + 1)) ∈ Fin)
48		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
49	48	elin2d 4161	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
50		prmnn 16757	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
51	49, 50	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
52	51	nnred 12266	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
53	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
54		zcn 12614	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
55	54	abscld 15516	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
56		peano2re 11401	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
57	55, 56	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℝ)
58	57	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℝ)
59		elinel1 4157	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
60		elicc2 13456	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
61	18, 11, 60	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
62	59, 61	imbitrid 247	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
63	62	imp 412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
64	63	simp3d 1162	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
65	54	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℂ)
66	65	abscld 15516	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
67	53	leabsd 15492	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
68	66	lep1d 12164	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
69	53, 66, 58, 67, 68	letrd 11385	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
70	52, 53, 58, 64, 69	letrd 11385	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
71		prmuz2 16779	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
72	49, 71	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
73		nn0abscl 15389	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ₀)
74		nn0p1nn 12561	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) ∈ ℕ₀ → ((abs‘𝐴) + 1) ∈ ℕ)
75	73, 74	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
76	75	nnzd 12635	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
77	76	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
78		elfz5 13562	. . . . . . . . 9 ⊢ ((𝑝 ∈ (ℤ_≥‘2) ∧ ((abs‘𝐴) + 1) ∈ ℤ) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
79	72, 77, 78	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
80	70, 79	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (2...((abs‘𝐴) + 1)))
81	80	ex 418	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (2...((abs‘𝐴) + 1))))
82	81	ssrdv 3946	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
83	47, 82	ssfid 9239	. . . 4 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
84		fzfid 14029	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
85		simprl 783	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
86	85	elin2d 4161	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
87		elfznn 13600	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
88	87	ad2antll 742	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
89		vmappw 27317	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
90	86, 88, 89	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
91	51	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
92	91	nnrpd 13076	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ⁺)
93	92	relogcld 26825	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
94	90, 93	eqeltrd 2866	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) ∈ ℝ)
95	88	nnnn0d 12583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ₀)
96		nnexpcl 14130	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑝↑𝑘) ∈ ℕ)
97	91, 95, 96	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℕ)
98	97	nnrpd 13076	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℝ⁺)
99	98	relogcld 26825	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝↑𝑘)) ∈ ℝ)
100		ifcl 4538	. . . . . . . . 9 ⊢ (((log‘(𝑝↑𝑘)) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) ∈ ℝ)
101	99, 18, 100	sylancl 598	. . . . . . . 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 11660	. . . . . . 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 12308	. . . . . 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 473	. . . . 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 15811	. . . 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 15811	. . 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 13076	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
108	107	relogcld 26825	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
109		uz2m1nn 12965	. . . . . . 7 ⊢ (𝑝 ∈ (ℤ_≥‘2) → (𝑝 − 1) ∈ ℕ)
110	72, 109	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℕ)
111	51, 110	nnmulcld 12307	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
112	108, 111	nndivred 12308	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
113	83, 112	fsumrecl 15811	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
114		2re 12333	. . . 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 12303	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
118	116, 52, 53, 117, 64	ltletrd 11388	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
119	53, 118	elrpd 13075	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
120	119	relogcld 26825	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
121		prmgt1 16781	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
122	49, 121	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
123	52, 122	rplogcld 26831	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
124	120, 123	rerpdivcld 13109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
125	123	rpcnd 13080	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
126	125	mullidd 11245	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
127	107, 119	logled 26829	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ≤ 𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
128	64, 127	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
129	126, 128	eqbrtrd 5138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
130		1re 11226	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
131	130	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
132	131, 120, 123	lemuldivd 13127	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
133	129, 132	mpbid 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
134		flge1nn 13874	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
135	124, 133, 134	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
136		nnuz 12919	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
137	135, 136	eleqtrdi 2876	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1))
138	103	recnd 11255	. . . . . . . 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 473	. . . . . . 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 7431	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑝↑𝑘) = (𝑝↑1))
141	140	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑘 = 1 → (Λ‘(𝑝↑𝑘)) = (Λ‘(𝑝↑1)))
142	140	eleq1d 2851	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑝↑𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
143	140	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (log‘(𝑝↑𝑘)) = (log‘(𝑝↑1)))
144	142, 143	ifbieq1d 4517	. . . . . . . . 9 ⊢ (𝑘 = 1 → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
145	141, 144	oveq12d 7441	. . . . . . . 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 7441	. . . . . . 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 15830	. . . . . 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 12267	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
149	148	exp1d 14197	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
150	149	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
151		vmaprm 27318	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → (Λ‘𝑝) = (log‘𝑝))
152	49, 151	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑝) = (log‘𝑝))
153	150, 152	eqtrd 2801	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
154	149, 49	eqeltrd 2866	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
155	154	iftrued 4500	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
156	149	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
157	155, 156	eqtrd 2801	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
158	153, 157	oveq12d 7441	. . . . . . . . . 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 11575	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
160	158, 159	eqtrd 2801	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
161	160, 149	oveq12d 7441	. . . . . . . 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 13087	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
163		div0 11921	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (0 / 𝑝) = 0)
164	162, 163	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 / 𝑝) = 0)
165	161, 164	eqtrd 2801	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
166		1p1e2 12382	. . . . . . . . . 10 ⊢ (1 + 1) = 2
167	166	oveq1i 7433	. . . . . . . . 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 13566	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
170		eluz2nn 12930	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
171	169, 170	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
172	171, 167	eleq2s 2884	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
173	49, 172, 89	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
174	51	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℕ)
175		nnq 13004	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ → 𝑝 ∈ ℚ)
176	174, 175	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℚ)
177	169, 167	eleq2s 2884	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
178	177	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑘 ∈ (ℤ_≥‘2))
179		expnprm 16987	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℚ ∧ 𝑘 ∈ (ℤ_≥‘2)) → ¬ (𝑝↑𝑘) ∈ ℙ)
180	176, 178, 179	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ¬ (𝑝↑𝑘) ∈ ℙ)
181	180	iffalsed 4503	. . . . . . . . . . 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 7441	. . . . . . . . . 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 11576	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − 0) = (log‘𝑝))
184	183	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) − 0) = (log‘𝑝))
185	182, 184	eqtrd 2801	. . . . . . . . 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 7438	. . . . . . . 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 15783	. . . . . . 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 7441	. . . . . 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 14029	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
190	108	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ)
191		nnnn0 12529	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
192	51, 191, 96	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℕ)
193	190, 192	nndivred 12308	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
194	171, 193	sylan2 605	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
195	189, 194	fsumrecl 15811	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
196	195	recnd 11255	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℂ)
197	196	addlidd 11429	. . . . . 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 2805	. . . . 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 13088	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ⁺)
200	124	flcld 13851	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
201	200	peano2zd 12721	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
202	199, 201	rpexpcld 14303	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ⁺)
203	202	rpge0d 13082	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
204	51	nnrecred 12305	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
205	204	resqcld 14181	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
206	135	peano2nnd 12268	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
207	206	nnnn0d 12583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ₀)
208	204, 207	reexpcld 14219	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
209	205, 208	subge02d 11824	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2))
211	110	nnrpd 13076	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ⁺)
212	211	rpcnne0d 13087	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
213	199	rpcnd 13080	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
214		dmdcan 11943	. . . . . . . . . . 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 1398	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
216	131	recnd 11255	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
217		divsubdir 11926	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
218	148, 216, 162, 217	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
219		divid 11920	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
220	162, 219	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 / 𝑝) = 1)
221	220	oveq1d 7438	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
222	218, 221	eqtrd 2801	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
223		divdiv1 11944	. . . . . . . . . . . 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 1398	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
225	222, 224	oveq12d 7441	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
226	51	nnne0d 12304	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
227	213, 148, 226	divrecd 12012	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
228	213	sqvald 14199	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
229	227, 228	eqtr4d 2804	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
230	215, 225, 229	3eqtr3d 2809	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
231	210, 230	breqtrrd 5144	. . . . . . . 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 11660	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
233	111	nnrecred 12305	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
234		resubcl 11540	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑝) ∈ ℝ) → (1 − (1 / 𝑝)) ∈ ℝ)
235	130, 204, 234	sylancr 599	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ)
236		recgt1 12129	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
237	52, 117, 236	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
238	122, 237	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) < 1)
239		posdif 11725	. . . . . . . . . . 11 ⊢ (((1 / 𝑝) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
240	204, 130, 239	sylancl 598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
241	238, 240	mpbid 235	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < (1 − (1 / 𝑝)))
242		ledivmul 12109	. . . . . . . . 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 1401	. . . . . . . 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 260	. . . . . . 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 13075	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ⁺)
246	232, 245	rerpdivcld 13109	. . . . . . . 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 13122	. . . . . . 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 235	. . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℂ)
250	192	nncnd 12267	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℂ)
251	192	nnne0d 12304	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ≠ 0)
252	249, 250, 251	divrecd 12012	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
253	148	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ)
254	51	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ)
255	254	nnne0d 12304	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
256		nnz 12630	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
257	256	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
258	253, 255, 257	exprecd 14210	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝↑𝑘)))
259	258	oveq2d 7439	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
260	252, 259	eqtr4d 2804	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
261	171, 260	sylan2 605	. . . . . . . 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 15779	. . . . . . 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 12583	. . . . . . . . 9 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ₀)
264		expcl 14135	. . . . . . . . 9 ⊢ (((1 / 𝑝) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
265	213, 263, 264	syl2an 608	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
266	189, 125, 265	fsummulc2 15861	. . . . . . 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 13782	. . . . . . . . . . 11 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
268	200, 267	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
269	268	sumeq1d 15777	. . . . . . . . 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 11364	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
271		2nn0 12539	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
272	271	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ₀)
273		eluzp1p1 12908	. . . . . . . . . . . 12 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
274	137, 273	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
275		df-2 12321	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
276	275	fveq2i 6891	. . . . . . . . . . 11 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
277	274, 276	eleqtrrdi 2877	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘2))
278	213, 270, 272, 277	geoserg 15946	. . . . . . . . 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 2801	. . . . . . . 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 7439	. . . . . . 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 2807	. . . . . 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 12267	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
283	111	nnne0d 12304	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
284	125, 282, 283	divrecd 12012	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
285	248, 281, 284	3brtr4d 5148	. . . . 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 5138	. . . 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 15877	. . 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 13566	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ_≥‘2))
289		eluz2nn 12930	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℕ)
290	288, 289	syl 18	. . . . . . . . . 10 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ ℕ)
291	290	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℕ)
292	291	nnred 12266	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℝ)
293	288	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ (ℤ_≥‘2))
294		eluz2gt1 12962	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
295	293, 294	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 1 < 𝑝)
296	292, 295	rplogcld 26831	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (log‘𝑝) ∈ ℝ⁺)
297	293, 109	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 − 1) ∈ ℕ)
298	291, 297	nnmulcld 12307	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
299	298	nnrpd 13076	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ⁺)
300	296, 299	rpdivcld 13095	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ⁺)
301	300	rpred 13078	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
302	47, 301	fsumrecl 15811	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
303	300	rpge0d 13082	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
304	47, 301, 303, 82	fsumless 15874	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
305		rplogsumlem1 27685	. . . . 5 ⊢ (((abs‘𝐴) + 1) ∈ ℕ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
306	75, 305	syl 18	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
307	113, 302, 115, 304, 306	letrd 11385	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
308	106, 113, 115, 287, 307	letrd 11385	. 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 5138	1 ⊢ (𝐴 ∈ ℤ → Σ𝑛 ∈ (1...𝐴)(((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ≤ 2)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∩ cin 3907 ifcif 4492 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 · cmul 11123 < clt 11261 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℕcn 12251 2c2 12313 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ℚcq 12990 ℝ⁺crp 13034 [,]cicc 13393 ...cfz 13553 ..^cfzo 13701 ⌊cfl 13843 ↑cexp 14117 abscabs 15311 Σcsu 15763 ℙcprime 16754 logclog 26756 Λcvma 27293

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-shft 15130 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-tan 16150 df-pi 16151 df-dvds 16336 df-gcd 16578 df-prm 16755 df-pc 16922 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-submnd 18873 df-mulg 19165 df-cntz 19418 df-cmn 19883 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-haus 23509 df-cmp 23581 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063 df-log 26758 df-cxp 26759 df-vma 27299

This theorem is used by: rplogsum 27728

W3C validator