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Theorem rplogsumlem2 25174

Description: Lemma for rplogsum 25216. 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 12609	. . . . 5 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴)
2	1	oveq2d 6666	. . . 4 ⊢ (𝐴 ∈ ℤ → (1...(⌊‘𝐴)) = (1...𝐴))
3	2	sumeq1d 14431	. . 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 6191	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → (Λ‘𝑛) = (Λ‘(𝑝↑𝑘)))
5		eleq1 2689	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (𝑛 ∈ ℙ ↔ (𝑝↑𝑘) ∈ ℙ))
6		fveq2 6191	. . . . . . 7 ⊢ (𝑛 = (𝑝↑𝑘) → (log‘𝑛) = (log‘(𝑝↑𝑘)))
7	5, 6	ifbieq1d 4109	. . . . . 6 ⊢ (𝑛 = (𝑝↑𝑘) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0))
8	4, 7	oveq12d 6668	. . . . 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 6668	. . . 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 11381	. . . 4 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
12		elfznn 12370	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
13	12	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
14		vmacl 24844	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) ∈ ℝ)
16	13	nnrpd 11870	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
17	16	relogcld 24369	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (log‘𝑛) ∈ ℝ)
18		0re 10040	. . . . . . . 8 ⊢ 0 ∈ ℝ
19		ifcl 4130	. . . . . . . 8 ⊢ (((log‘𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
20	17, 18, 19	sylancl 694	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) ∈ ℝ)
21	15, 20	resubcld 10458	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) ∈ ℝ)
22	21, 13	nndivred 11069	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℝ)
23	22	recnd 10068	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) ∈ ℂ)
24		simprr 796	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (Λ‘𝑛) = 0)
25		vmaprm 24843	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) = (log‘𝑛))
26		prmnn 15388	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ)
27	26	nnred 11035	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℝ)
28		prmgt1 15409	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℙ → 1 < 𝑛)
29	27, 28	rplogcld 24375	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℙ → (log‘𝑛) ∈ ℝ⁺)
30	25, 29	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ∈ ℝ⁺)
31	30	rpne0d 11877	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℙ → (Λ‘𝑛) ≠ 0)
32	31	necon2bi 2824	. . . . . . . . . 10 ⊢ ((Λ‘𝑛) = 0 → ¬ 𝑛 ∈ ℙ)
33	32	ad2antll 765	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ¬ 𝑛 ∈ ℙ)
34	33	iffalsed 4097	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → if(𝑛 ∈ ℙ, (log‘𝑛), 0) = 0)
35	24, 34	oveq12d 6668	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = (0 − 0))
36		0m0e0 11130	. . . . . . 7 ⊢ (0 − 0) = 0
37	35, 36	syl6eq 2672	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → ((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) = 0)
38	37	oveq1d 6665	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (((Λ‘𝑛) − if(𝑛 ∈ ℙ, (log‘𝑛), 0)) / 𝑛) = (0 / 𝑛))
39	12	ad2antrl 764	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℕ)
40	39	nnrpd 11870	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → 𝑛 ∈ ℝ⁺)
41	40	rpcnne0d 11881	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑛) = 0)) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
42		div0 10715	. . . . . 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 2656	. . . 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 24939	. . 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 2658	. 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 12772	. . . . 5 ⊢ (𝐴 ∈ ℤ → (2...((abs‘𝐴) + 1)) ∈ Fin)
48		inss2 3834	. . . . . . . . . . . 12 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ ℙ
49		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
50	48, 49	sseldi 3601	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ)
51		prmnn 15388	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
52	50, 51	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ)
53	52	nnred 11035	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ)
54	11	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ)
55		zcn 11382	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
56	55	abscld 14175	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℝ)
57		peano2re 10209	. . . . . . . . . . 11 ⊢ ((abs‘𝐴) ∈ ℝ → ((abs‘𝐴) + 1) ∈ ℝ)
58	56, 57	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℝ)
59	58	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℝ)
60		inss1 3833	. . . . . . . . . . . . 13 ⊢ ((0[,]𝐴) ∩ ℙ) ⊆ (0[,]𝐴)
61	60	sseli 3599	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (0[,]𝐴))
62		elicc2 12238	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
63	18, 11, 62	sylancr 695	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
64	61, 63	syl5ib 234	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
65	64	imp 445	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))
66	65	simp3d 1075	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴)
67	55	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℂ)
68	67	abscld 14175	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ∈ ℝ)
69	54	leabsd 14153	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ (abs‘𝐴))
70	68	lep1d 10955	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (abs‘𝐴) ≤ ((abs‘𝐴) + 1))
71	54, 68, 59, 69, 70	letrd 10194	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ≤ ((abs‘𝐴) + 1))
72	53, 54, 59, 66, 71	letrd 10194	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ ((abs‘𝐴) + 1))
73		prmuz2 15408	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
74	50, 73	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (ℤ_≥‘2))
75		nn0abscl 14052	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℤ → (abs‘𝐴) ∈ ℕ₀)
76		nn0p1nn 11332	. . . . . . . . . . . 12 ⊢ ((abs‘𝐴) ∈ ℕ₀ → ((abs‘𝐴) + 1) ∈ ℕ)
77	75, 76	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℕ)
78	77	nnzd 11481	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → ((abs‘𝐴) + 1) ∈ ℤ)
79	78	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((abs‘𝐴) + 1) ∈ ℤ)
80		elfz5 12334	. . . . . . . . 9 ⊢ ((𝑝 ∈ (ℤ_≥‘2) ∧ ((abs‘𝐴) + 1) ∈ ℤ) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
81	74, 79, 80	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (2...((abs‘𝐴) + 1)) ↔ 𝑝 ≤ ((abs‘𝐴) + 1)))
82	72, 81	mpbird 247	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (2...((abs‘𝐴) + 1)))
83	82	ex 450	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ (2...((abs‘𝐴) + 1))))
84	83	ssrdv 3609	. . . . 5 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1)))
85		ssfi 8180	. . . . 5 ⊢ (((2...((abs‘𝐴) + 1)) ∈ Fin ∧ ((0[,]𝐴) ∩ ℙ) ⊆ (2...((abs‘𝐴) + 1))) → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
86	47, 84, 85	syl2anc 693	. . . 4 ⊢ (𝐴 ∈ ℤ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
87		fzfid 12772	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
88		simprl 794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ))
89	48, 88	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℙ)
90		elfznn 12370	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
91	90	ad2antll 765	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ)
92		vmappw 24842	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
93	89, 91, 92	syl2anc 693	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
94	52	adantrr 753	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℕ)
95	94	nnrpd 11870	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑝 ∈ ℝ⁺)
96	95	relogcld 24369	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈ ℝ)
97	93, 96	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (Λ‘(𝑝↑𝑘)) ∈ ℝ)
98	91	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → 𝑘 ∈ ℕ₀)
99		nnexpcl 12873	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ ∧ 𝑘 ∈ ℕ₀) → (𝑝↑𝑘) ∈ ℕ)
100	94, 98, 99	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℕ)
101	100	nnrpd 11870	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (𝑝↑𝑘) ∈ ℝ⁺)
102	101	relogcld 24369	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘(𝑝↑𝑘)) ∈ ℝ)
103		ifcl 4130	. . . . . . . . 9 ⊢ (((log‘(𝑝↑𝑘)) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) ∈ ℝ)
104	102, 18, 103	sylancl 694	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) ∈ ℝ)
105	97, 104	resubcld 10458	. . . . . . 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)
106	105, 100	nndivred 11069	. . . . . 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)
107	106	anassrs 680	. . . . 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)
108	87, 107	fsumrecl 14465	. . . 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)
109	86, 108	fsumrecl 14465	. . 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)
110	52	nnrpd 11870	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
111	110	relogcld 24369	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
112		uz2m1nn 11763	. . . . . . 7 ⊢ (𝑝 ∈ (ℤ_≥‘2) → (𝑝 − 1) ∈ ℕ)
113	74, 112	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℕ)
114	52, 113	nnmulcld 11068	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
115	111, 114	nndivred 11069	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
116	86, 115	fsumrecl 14465	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
117		2re 11090	. . . 4 ⊢ 2 ∈ ℝ
118	117	a1i 11	. . 3 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℝ)
119	18	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈ ℝ)
120	52	nngt0d 11064	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝)
121	119, 53, 54, 120, 66	ltletrd 10197	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴)
122	54, 121	elrpd 11869	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ⁺)
123	122	relogcld 24369	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈ ℝ)
124		prmgt1 15409	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 1 < 𝑝)
125	50, 124	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝)
126	53, 125	rplogcld 24375	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℝ⁺)
127	123, 126	rerpdivcld 11903	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
128	126	rpcnd 11874	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
129	128	mulid2d 10058	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) = (log‘𝑝))
130	110, 122	logled 24373	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ≤ 𝐴 ↔ (log‘𝑝) ≤ (log‘𝐴)))
131	66, 130	mpbid 222	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ≤ (log‘𝐴))
132	129, 131	eqbrtrd 4675	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 · (log‘𝑝)) ≤ (log‘𝐴))
133		1re 10039	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
134	133	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℝ)
135	134, 123, 126	lemuldivd 11921	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 1 ≤ ((log‘𝐴) / (log‘𝑝))))
136	132, 135	mpbid 222	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ≤ ((log‘𝐴) / (log‘𝑝)))
137		flge1nn 12622	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 1 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
138	127, 136, 137	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ)
139		nnuz 11723	. . . . . . . 8 ⊢ ℕ = (ℤ_≥‘1)
140	138, 139	syl6eleq 2711	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1))
141	106	recnd 10068	. . . . . . . 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)
142	141	anassrs 680	. . . . . . 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)
143		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑝↑𝑘) = (𝑝↑1))
144	143	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑘 = 1 → (Λ‘(𝑝↑𝑘)) = (Λ‘(𝑝↑1)))
145	143	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝑘 = 1 → ((𝑝↑𝑘) ∈ ℙ ↔ (𝑝↑1) ∈ ℙ))
146	143	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (log‘(𝑝↑𝑘)) = (log‘(𝑝↑1)))
147	145, 146	ifbieq1d 4109	. . . . . . . . 9 ⊢ (𝑘 = 1 → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0))
148	144, 147	oveq12d 6668	. . . . . . . 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)))
149	148, 143	oveq12d 6668	. . . . . . 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)))
150	140, 142, 149	fsum1p 14482	. . . . . 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))))
151	52	nncnd 11036	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℂ)
152	151	exp1d 13003	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) = 𝑝)
153	152	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (Λ‘𝑝))
154		vmaprm 24843	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℙ → (Λ‘𝑝) = (log‘𝑝))
155	50, 154	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘𝑝) = (log‘𝑝))
156	153, 155	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (Λ‘(𝑝↑1)) = (log‘𝑝))
157	152, 50	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝↑1) ∈ ℙ)
158	157	iftrued 4094	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘(𝑝↑1)))
159	152	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘(𝑝↑1)) = (log‘𝑝))
160	158, 159	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0) = (log‘𝑝))
161	156, 160	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = ((log‘𝑝) − (log‘𝑝)))
162	128	subidd 10380	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − (log‘𝑝)) = 0)
163	161, 162	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) = 0)
164	163, 152	oveq12d 6668	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = (0 / 𝑝))
165	110	rpcnne0d 11881	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0))
166		div0 10715	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (0 / 𝑝) = 0)
167	165, 166	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (0 / 𝑝) = 0)
168	164, 167	eqtrd 2656	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((Λ‘(𝑝↑1)) − if((𝑝↑1) ∈ ℙ, (log‘(𝑝↑1)), 0)) / (𝑝↑1)) = 0)
169		1p1e2 11134	. . . . . . . . . 10 ⊢ (1 + 1) = 2
170	169	oveq1i 6660	. . . . . . . . 9 ⊢ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝))))
171	170	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2...(⌊‘((log‘𝐴) / (log‘𝑝)))))
172		elfzuz 12338	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
173		eluz2nn 11726	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℕ)
174	172, 173	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
175	174, 170	eleq2s 2719	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
176	50, 175, 92	syl2an 494	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (Λ‘(𝑝↑𝑘)) = (log‘𝑝))
177	52	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℕ)
178		nnq 11801	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℕ → 𝑝 ∈ ℚ)
179	177, 178	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑝 ∈ ℚ)
180	172, 170	eleq2s 2719	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ (ℤ_≥‘2))
181	180	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 𝑘 ∈ (ℤ_≥‘2))
182		expnprm 15606	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℚ ∧ 𝑘 ∈ (ℤ_≥‘2)) → ¬ (𝑝↑𝑘) ∈ ℙ)
183	179, 181, 182	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ¬ (𝑝↑𝑘) ∈ ℙ)
184	183	iffalsed 4097	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → if((𝑝↑𝑘) ∈ ℙ, (log‘(𝑝↑𝑘)), 0) = 0)
185	176, 184	oveq12d 6668	. . . . . . . . . 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))
186	128	subid1d 10381	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) − 0) = (log‘𝑝))
187	186	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ((1 + 1)...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) − 0) = (log‘𝑝))
188	185, 187	eqtrd 2656	. . . . . . . . 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))
189	188	oveq1d 6665	. . . . . . . 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)))
190	171, 189	sumeq12dv 14437	. . . . . . 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)))
191	168, 190	oveq12d 6668	. . . . . 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))))
192		fzfid 12772	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin)
193	111	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℝ)
194		nnnn0 11299	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
195	52, 194, 99	syl2an 494	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℕ)
196	193, 195	nndivred 11069	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
197	174, 196	sylan2 491	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
198	192, 197	fsumrecl 14465	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℝ)
199	198	recnd 10068	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ∈ ℂ)
200	199	addid2d 10237	. . . . . 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)))
201	150, 191, 200	3eqtrd 2660	. . . . 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)))
202	110	rpreccld 11882	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ⁺)
203	127	flcld 12599	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
204	203	peano2zd 11485	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℤ)
205	202, 204	rpexpcld 13032	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ⁺)
206	205	rpge0d 11876	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤ ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
207	52	nnrecred 11066	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℝ)
208	207	resqcld 13035	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) ∈ ℝ)
209	138	peano2nnd 11037	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ)
210	209	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ ℕ₀)
211	207, 210	reexpcld 13025	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)) ∈ ℝ)
212	208, 211	subge02d 10619	. . . . . . . . . 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)))
213	206, 212	mpbid 222	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 / 𝑝)↑2))
214	113	nnrpd 11870	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 − 1) ∈ ℝ⁺)
215	214	rpcnne0d 11881	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0))
216	202	rpcnd 11874	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ∈ ℂ)
217		dmdcan 10735	. . . . . . . . . . 11 ⊢ ((((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0) ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ (1 / 𝑝) ∈ ℂ) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
218	215, 165, 216, 217	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 / 𝑝) / 𝑝))
219	134	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈ ℂ)
220		divsubdir 10721	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
221	151, 219, 165, 220	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = ((𝑝 / 𝑝) − (1 / 𝑝)))
222		divid 10714	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) → (𝑝 / 𝑝) = 1)
223	165, 222	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 / 𝑝) = 1)
224	223	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 / 𝑝) − (1 / 𝑝)) = (1 − (1 / 𝑝)))
225	221, 224	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((𝑝 − 1) / 𝑝) = (1 − (1 / 𝑝)))
226		divdiv1 10736	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ (𝑝 ∈ ℂ ∧ 𝑝 ≠ 0) ∧ ((𝑝 − 1) ∈ ℂ ∧ (𝑝 − 1) ≠ 0)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
227	219, 165, 215, 226	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / (𝑝 − 1)) = (1 / (𝑝 · (𝑝 − 1))))
228	225, 227	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((𝑝 − 1) / 𝑝) · ((1 / 𝑝) / (𝑝 − 1))) = ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
229	52	nnne0d 11065	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≠ 0)
230	216, 151, 229	divrecd 10804	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝) · (1 / 𝑝)))
231	216	sqvald 13005	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝)↑2) = ((1 / 𝑝) · (1 / 𝑝)))
232	230, 231	eqtr4d 2659	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) / 𝑝) = ((1 / 𝑝)↑2))
233	218, 228, 232	3eqtr3d 2664	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))) = ((1 / 𝑝)↑2))
234	213, 233	breqtrrd 4681	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ≤ ((1 − (1 / 𝑝)) · (1 / (𝑝 · (𝑝 − 1)))))
235	208, 211	resubcld 10458	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) ∈ ℝ)
236	114	nnrecred 11066	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / (𝑝 · (𝑝 − 1))) ∈ ℝ)
237		resubcl 10345	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (1 / 𝑝) ∈ ℝ) → (1 − (1 / 𝑝)) ∈ ℝ)
238	133, 207, 237	sylancr 695	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ)
239		recgt1 10919	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℝ ∧ 0 < 𝑝) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
240	53, 120, 239	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 < 𝑝 ↔ (1 / 𝑝) < 1))
241	125, 240	mpbid 222	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) < 1)
242		posdif 10521	. . . . . . . . . . 11 ⊢ (((1 / 𝑝) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
243	207, 133, 242	sylancl 694	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((1 / 𝑝) < 1 ↔ 0 < (1 − (1 / 𝑝))))
244	241, 243	mpbid 222	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < (1 − (1 / 𝑝)))
245		ledivmul 10899	. . . . . . . . 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))))))
246	235, 236, 238, 244, 245	syl112anc 1330	. . . . . . . 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))))))
247	234, 246	mpbird 247	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ≤ (1 / (𝑝 · (𝑝 − 1))))
248	238, 244	elrpd 11869	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 − (1 / 𝑝)) ∈ ℝ⁺)
249	235, 248	rerpdivcld 11903	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((((1 / 𝑝)↑2) − ((1 / 𝑝)↑((⌊‘((log‘𝐴) / (log‘𝑝))) + 1))) / (1 − (1 / 𝑝))) ∈ ℝ)
250	249, 236, 126	lemul2d 11916	. . . . . . 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))))))
251	247, 250	mpbid 222	. . . . . 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)))))
252	128	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (log‘𝑝) ∈ ℂ)
253	195	nncnd 11036	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ∈ ℂ)
254	195	nnne0d 11065	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → (𝑝↑𝑘) ≠ 0)
255	252, 253, 254	divrecd 10804	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
256	151	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℂ)
257	52	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ∈ ℕ)
258	257	nnne0d 11065	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑝 ≠ 0)
259		nnz 11399	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
260	259	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
261	256, 258, 260	exprecd 13016	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑝)↑𝑘) = (1 / (𝑝↑𝑘)))
262	261	oveq2d 6666	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) · ((1 / 𝑝)↑𝑘)) = ((log‘𝑝) · (1 / (𝑝↑𝑘))))
263	255, 262	eqtr4d 2659	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ ℕ) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
264	174, 263	sylan2 491	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((log‘𝑝) / (𝑝↑𝑘)) = ((log‘𝑝) · ((1 / 𝑝)↑𝑘)))
265	264	sumeq2dv 14433	. . . . . . 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)))
266	174	nnnn0d 11351	. . . . . . . . 9 ⊢ (𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ₀)
267		expcl 12878	. . . . . . . . 9 ⊢ (((1 / 𝑝) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
268	216, 266, 267	syl2an 494	. . . . . . . 8 ⊢ (((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) ∧ 𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((1 / 𝑝)↑𝑘) ∈ ℂ)
269	192, 128, 268	fsummulc2 14516	. . . . . . 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)))
270		fzval3 12536	. . . . . . . . . . 11 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
271	203, 270	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (2...(⌊‘((log‘𝐴) / (log‘𝑝)))) = (2..^((⌊‘((log‘𝐴) / (log‘𝑝))) + 1)))
272	271	sumeq1d 14431	. . . . . . . . 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))
273	207, 241	ltned 10173	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (1 / 𝑝) ≠ 1)
274		2nn0 11309	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
275	274	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 2 ∈ ℕ₀)
276		eluzp1p1 11713	. . . . . . . . . . . 12 ⊢ ((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ (ℤ_≥‘1) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
277	140, 276	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘(1 + 1)))
278		df-2 11079	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
279	278	fveq2i 6194	. . . . . . . . . . 11 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
280	277, 279	syl6eleqr 2712	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((⌊‘((log‘𝐴) / (log‘𝑝))) + 1) ∈ (ℤ_≥‘2))
281	216, 273, 275, 280	geoserg 14598	. . . . . . . . 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))))
282	272, 281	eqtrd 2656	. . . . . . . 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))))
283	282	oveq2d 6666	. . . . . . 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)))))
284	265, 269, 283	3eqtr2d 2662	. . . . . 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)))))
285	114	nncnd 11036	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ∈ ℂ)
286	114	nnne0d 11065	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 · (𝑝 − 1)) ≠ 0)
287	128, 285, 286	divrecd 10804	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) = ((log‘𝑝) · (1 / (𝑝 · (𝑝 − 1)))))
288	251, 284, 287	3brtr4d 4685	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈ (2...(⌊‘((log‘𝐴) / (log‘𝑝))))((log‘𝑝) / (𝑝↑𝑘)) ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
289	201, 288	eqbrtrd 4675	. . . 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))))
290	86, 108, 115, 289	fsumle 14531	. . 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))))
291		elfzuz 12338	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ (ℤ_≥‘2))
292		eluz2nn 11726	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℕ)
293	291, 292	syl 17	. . . . . . . . . 10 ⊢ (𝑝 ∈ (2...((abs‘𝐴) + 1)) → 𝑝 ∈ ℕ)
294	293	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℕ)
295	294	nnred 11035	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ ℝ)
296	291	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 𝑝 ∈ (ℤ_≥‘2))
297		eluz2b2 11761	. . . . . . . . . 10 ⊢ (𝑝 ∈ (ℤ_≥‘2) ↔ (𝑝 ∈ ℕ ∧ 1 < 𝑝))
298	297	simprbi 480	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
299	296, 298	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 1 < 𝑝)
300	295, 299	rplogcld 24375	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (log‘𝑝) ∈ ℝ⁺)
301	296, 112	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 − 1) ∈ ℕ)
302	294, 301	nnmulcld 11068	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℕ)
303	302	nnrpd 11870	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → (𝑝 · (𝑝 − 1)) ∈ ℝ⁺)
304	300, 303	rpdivcld 11889	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ⁺)
305	304	rpred 11872	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → ((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
306	47, 305	fsumrecl 14465	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ∈ ℝ)
307	304	rpge0d 11876	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑝 ∈ (2...((abs‘𝐴) + 1))) → 0 ≤ ((log‘𝑝) / (𝑝 · (𝑝 − 1))))
308	47, 305, 307, 84	fsumless 14528	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))))
309		rplogsumlem1 25173	. . . . 5 ⊢ (((abs‘𝐴) + 1) ∈ ℕ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
310	77, 309	syl 17	. . . 4 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ (2...((abs‘𝐴) + 1))((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
311	116, 306, 118, 308, 310	letrd 10194	. . 3 ⊢ (𝐴 ∈ ℤ → Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) / (𝑝 · (𝑝 − 1))) ≤ 2)
312	109, 116, 118, 290, 311	letrd 10194	. 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)
313	46, 312	eqbrtrd 4675	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℚcq 11788 ℝ⁺crp 11832 [,]cicc 12178 ...cfz 12326 ..^cfzo 12465 ⌊cfl 12591 ↑cexp 12860 abscabs 13974 Σcsu 14416 ℙcprime 15385 logclog 24301 Λcvma 24818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-vma 24824

This theorem is referenced by: rplogsum 25216

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