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Theorem rplogsumlem2 27452

Description: Lemma for rplogsum 27494. 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 13728	. . . . 5 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2	1	oveq2d 7374	. . . 4 ⊢ (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
3	2	sumeq1d 15623	. . 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 6834	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → (Λ‘𝑛) = (Λ‘(𝑝↑𝑘)))
5		eleq1 2824	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (𝑛 ∈ ℙ ↔ (𝑝↑𝑘) ∈ ℙ))
6		fveq2 6834	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (log‘𝑛) = (log‘(𝑝↑𝑘)))
7	5, 6	ifbieq1d 4504	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0))
8	4, 7	oveq12d 7376	. . . . 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 7376	. . . 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 12492	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12		elfznn 13469	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14		vmacl 27084	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
16	13	nnrpd 12947	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
17	16	relogcld 26588	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18		0re 11134	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4525	. . . . . . . 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 11565	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
22	21, 13	nndivred 12199	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
23	22	recnd 11160	. . . 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 27083	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26		prmnn 16601	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	nnred 12160	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28		prmgt1 16624	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 1 < 𝑛)
29	27, 28	rplogcld 26594	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ⁺)
30	25, 29	eqeltrd 2836	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ⁺)
31	30	rpne0d 12954	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
32	31	necon2bi 2962	. . . . . . . . . 10 ⊢ ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
33	32	ad2antll 729	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4490	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
35	24, 34	oveq12d 7376	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36		0m0e0 12260	. . . . . . 7 ⊢ (0 − 0) = 0
37	35, 36	eqtrdi 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
38	37	oveq1d 7373	. . . . 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 12947	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ⁺)
41	40	rpcnne0d 12958	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42		div0 11829	. . . . . 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 2771	. . . 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 27181	. . 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 2773	. 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 13896	. . . . 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 4157	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
50		prmnn 16601	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
52	51	nnred 12160	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
53	11	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
54		zcn 12493	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
55	54	abscld 15362	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
56		peano2re 11306	. . . . . . . . . . 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 4153	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
60		elicc2 13327	. . . . . . . . . . . . 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 15362	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
67	53	leabsd 15338	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
68	66	lep1d 12073	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
69	53, 66, 58, 67, 68	letrd 11290	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
70	52, 53, 58, 64, 69	letrd 11290	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
71		prmuz2 16623	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
72	49, 71	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
73		nn0abscl 15235	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ₀)
74		nn0p1nn 12440	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) ∈ ℕ₀ → ((abs‘𝐴) + 1) ∈ ℕ)
75	73, 74	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
76	75	nnzd 12514	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
77	76	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
78		elfz5 13432	. . . . . . . . 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 3939	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
83	47, 82	ssfid 9169	. . . 4 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
84		fzfid 13896	. . . . 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 4157	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
87		elfznn 13469	. . . . . . . . . . 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 27082	. . . . . . . . . 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 12947	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ⁺)
93	92	relogcld 26588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
94	90, 93	eqeltrd 2836	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) ∈ ℝ)
95	88	nnnn0d 12462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ₀)
96		nnexpcl 13997	. . . . . . . . . . . 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 12947	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℝ⁺)
99	98	relogcld 26588	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝↑𝑘)) ∈ ℝ)
100		ifcl 4525	. . . . . . . . 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 11565	. . . . . . 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 12199	. . . . . 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 15657	. . . 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 15657	. . 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 12947	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
108	107	relogcld 26588	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
109		uz2m1nn 12836	. . . . . . 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 12198	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
112	108, 111	nndivred 12199	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
113	83, 112	fsumrecl 15657	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
114		2re 12219	. . . 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 12194	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
118	116, 52, 53, 117, 64	ltletrd 11293	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
119	53, 118	elrpd 12946	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
120	119	relogcld 26588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
121		prmgt1 16624	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
122	49, 121	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
123	52, 122	rplogcld 26594	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
124	120, 123	rerpdivcld 12980	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
125	123	rpcnd 12951	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
126	125	mullidd 11150	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
127	107, 119	logled 26592	. . . . . . . . . . . 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 5120	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
130		1re 11132	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
131	130	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
132	131, 120, 123	lemuldivd 12998	. . . . . . . . . 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 13741	. . . . . . . . 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 12790	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
137	135, 136	eleqtrdi 2846	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1))
138	103	recnd 11160	. . . . . . . 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 7366	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑝↑𝑘) = (𝑝↑1))
141	140	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑘 = 1 → (Λ‘(𝑝↑𝑘)) = (Λ‘(𝑝↑1)))
142	140	eleq1d 2821	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑝↑𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
143	140	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (log‘(𝑝↑𝑘)) = (log‘(𝑝↑1)))
144	142, 143	ifbieq1d 4504	. . . . . . . . 9 ⊢ (𝑘 = 1 → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
145	141, 144	oveq12d 7376	. . . . . . . 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 7376	. . . . . . 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 15676	. . . . . 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 12161	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
149	148	exp1d 14064	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
150	149	fveq2d 6838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
151		vmaprm 27083	. . . . . . . . . . . . 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 2771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
154	149, 49	eqeltrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
155	154	iftrued 4487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
156	149	fveq2d 6838	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
157	155, 156	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
158	153, 157	oveq12d 7376	. . . . . . . . . 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 11480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
160	158, 159	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
161	160, 149	oveq12d 7376	. . . . . . . 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 12958	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
163		div0 11829	. . . . . . . . 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 2771	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
166		1p1e2 12265	. . . . . . . . . 10 ⊢ (1 + 1) = 2
167	166	oveq1i 7368	. . . . . . . . 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 13436	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
170		eluz2nn 12801	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
171	169, 170	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
172	171, 167	eleq2s 2854	. . . . . . . . . . . 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 12875	. . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . 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 16830	. . . . . . . . . . . . 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 4490	. . . . . . . . . . 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 7376	. . . . . . . . . 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 11481	. . . . . . . . . . 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 2771	. . . . . . . . 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 7373	. . . . . . . 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 15629	. . . . . . 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 7376	. . . . . 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 13896	. . . . . . . . 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 12408	. . . . . . . . . . . 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 12199	. . . . . . . . . 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 15657	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
196	195	recnd 11160	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℂ)
197	196	addlidd 11334	. . . . . 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 2775	. . . . 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 12959	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ⁺)
200	124	flcld 13718	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
201	200	peano2zd 12599	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
202	199, 201	rpexpcld 14170	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ⁺)
203	202	rpge0d 12953	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
204	51	nnrecred 12196	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
205	204	resqcld 14048	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
206	135	peano2nnd 12162	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
207	206	nnnn0d 12462	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ₀)
208	204, 207	reexpcld 14086	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
209	205, 208	subge02d 11729	. . . . . . . . . 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 12947	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ⁺)
212	211	rpcnne0d 12958	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
213	199	rpcnd 12951	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
214		dmdcan 11851	. . . . . . . . . . 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 11160	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
217		divsubdir 11835	. . . . . . . . . . . . 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 11827	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
222	218, 221	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
223		divdiv1 11852	. . . . . . . . . . . 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 7376	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
226	51	nnne0d 12195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
227	213, 148, 226	divrecd 11920	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
228	213	sqvald 14066	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
229	227, 228	eqtr4d 2774	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
230	215, 225, 229	3eqtr3d 2779	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
231	210, 230	breqtrrd 5126	. . . . . . . 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 11565	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
233	111	nnrecred 12196	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
234		resubcl 11445	. . . . . . . . . 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 12038	. . . . . . . . . . . 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 11630	. . . . . . . . . . 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 12018	. . . . . . . . 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 12946	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ⁺)
246	232, 245	rerpdivcld 12980	. . . . . . . 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 12993	. . . . . . 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 12161	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℂ)
251	192	nnne0d 12195	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ≠ 0)
252	249, 250, 251	divrecd 11920	. . . . . . . . . 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 12195	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
256		nnz 12509	. . . . . . . . . . . . 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 14077	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝↑𝑘)))
259	258	oveq2d 7374	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
260	252, 259	eqtr4d 2774	. . . . . . . . 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 15625	. . . . . . 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 12462	. . . . . . . . 9 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ₀)
264		expcl 14002	. . . . . . . . 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 15707	. . . . . . 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 13650	. . . . . . . . . . 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 15623	. . . . . . . . 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 11269	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
271		2nn0 12418	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
272	271	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ₀)
273		eluzp1p1 12779	. . . . . . . . . . . 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 12208	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
276	275	fveq2i 6837	. . . . . . . . . . 11 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
277	274, 276	eleqtrrdi 2847	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘2))
278	213, 270, 272, 277	geoserg 15789	. . . . . . . . 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 2771	. . . . . . . 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 7374	. . . . . . 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 2777	. . . . . 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 12161	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
283	111	nnne0d 12195	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
284	125, 282, 283	divrecd 11920	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
285	248, 281, 284	3brtr4d 5130	. . . . 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 5120	. . . 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 15722	. . 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 13436	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ_≥‘2))
289		eluz2nn 12801	. . . . . . . . . . 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 12160	. . . . . . . 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 12833	. . . . . . . . 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 26594	. . . . . . 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 12198	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
299	298	nnrpd 12947	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ⁺)
300	296, 299	rpdivcld 12966	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ⁺)
301	300	rpred 12949	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
302	47, 301	fsumrecl 15657	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
303	300	rpge0d 12953	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
304	47, 301, 303, 82	fsumless 15719	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
305		rplogsumlem1 27451	. . . . 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 11290	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
308	106, 113, 115, 287, 307	letrd 11290	. 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 5120	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 1541 ∈ wcel 2113 ≠ wne 2932 ∩ cin 3900 ifcif 4479 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ℚcq 12861 ℝ⁺crp 12905 [,]cicc 13264 ...cfz 13423 ..^cfzo 13570 ⌊cfl 13710 ↑cexp 13984 abscabs 15157 Σcsu 15609 ℙcprime 16598 logclog 26519 Λcvma 27058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-cxp 26522 df-vma 27064

This theorem is referenced by: rplogsum 27494

W3C validator