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Theorem fsumvma2 27131

Description: Apply fsumvma 27130 for the common case of all numbers less than a real number 𝐴. (Contributed by Mario Carneiro, 30-Apr-2016.)

**Hypotheses**
Ref	Expression
fsumvma2.1	⊢ (𝑥 = (𝑝↑𝑘) → 𝐵 = 𝐶)
fsumvma2.2	⊢ (𝜑 → 𝐴 ∈ ℝ)
fsumvma2.3	⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
fsumvma2.4	⊢ ((𝜑 ∧ (𝑥 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)

**Assertion**
Ref	Expression
fsumvma2	⊢ (𝜑 → Σ𝑥 ∈ (1...(⌊‘𝐴))𝐵 = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))𝐶)

Distinct variable groups: 𝑘,𝑝,𝑥,𝐴 𝑥,𝐶 𝜑,𝑘,𝑝,𝑥 𝐵,𝑘,𝑝 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑘,𝑝)

**Proof of Theorem fsumvma2**
Step	Hyp	Ref	Expression
1		fsumvma2.1	. 2 ⊢ (𝑥 = (𝑝↑𝑘) → 𝐵 = 𝐶)
2		fzfid 13944	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
3		fz1ssnn 13522	. . 3 ⊢ (1...(⌊‘𝐴)) ⊆ ℕ
4	3	a1i 11	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆ ℕ)
5		fsumvma2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ppifi 27022	. . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8		elinel2 4167	. . . . 5 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
9		elfznn 13520	. . . . 5 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
10	8, 9	anim12i 613	. . . 4 ⊢ ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
11	10	pm4.71ri 560	. . 3 ⊢ ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))))
12	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝐴 ∈ ℝ)
13		prmnn 16650	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	13	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ)
15		nnnn0 12455	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
16	15	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ₀)
17	14, 16	nnexpcld 14216	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℕ)
18	17	nnzd 12562	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℤ)
19		flge 13773	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℤ) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
20	12, 18, 19	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
21		simplrl 776	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℙ)
22	21, 13	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℕ)
23	22	nnrpd 12999	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℝ⁺)
24		simplrr 777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℕ)
25	24	nnzd 12562	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℤ)
26		relogexp 26511	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
27	23, 25, 26	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
28	27	breq1d 5119	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
29	24	nnred 12202	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℝ)
30	12	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ)
31		0red 11183	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 ∈ ℝ)
32	14	nnred 12202	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℝ)
33	32	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℝ)
34	22	nngt0d 12236	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝑝)
35		0red 11183	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ∈ ℝ)
36	14	nnnn0d 12509	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ₀)
37	36	nn0ge0d 12512	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ≤ 𝑝)
38		elicc2 13378	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
39		df-3an 1088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
40	38, 39	bitrdi 287	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
41	40	baibd 539	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
42	35, 12, 32, 37, 41	syl22anc 838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
43	42	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ≤ 𝐴)
44	31, 33, 30, 34, 43	ltletrd 11340	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝐴)
45	30, 44	elrpd 12998	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ⁺)
46	45	relogcld 26538	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝐴) ∈ ℝ)
47		prmuz2 16672	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
48		eluzelre 12810	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℝ)
49		eluz2gt1 12885	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
50	48, 49	rplogcld 26544	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘2) → (log‘𝑝) ∈ ℝ⁺)
51	21, 47, 50	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝑝) ∈ ℝ⁺)
52	29, 46, 51	lemuldivd 13050	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
53	46, 51	rerpdivcld 13032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
54		flge 13773	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
55	53, 25, 54	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
56	28, 52, 55	3bitrd 305	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
57	17	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℕ)
58	57	nnrpd 12999	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℝ⁺)
59	58, 45	logled 26542	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
60		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
61		nnuz 12842	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
62	60, 61	eleqtrdi 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ (ℤ_≥‘1))
63	62	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ (ℤ_≥‘1))
64	53	flcld 13766	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
65		elfz5 13483	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ_≥‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
66	63, 64, 65	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
67	56, 59, 66	3bitr4d 311	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))))
68	67	pm5.32da 579	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ (0[,]𝐴) ∧ (𝑝↑𝑘) ≤ 𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))))
69	14	nncnd 12203	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℂ)
70	69	exp1d 14112	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) = 𝑝)
71	14	nnge1d 12235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 1 ≤ 𝑝)
72	32, 71, 62	leexp2ad 14225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) ≤ (𝑝↑𝑘))
73	70, 72	eqbrtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ≤ (𝑝↑𝑘))
74	17	nnred 12202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℝ)
75		letr 11274	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
76	32, 74, 12, 75	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
77	73, 76	mpand 695	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
78	77, 42	sylibrd 259	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ∈ (0[,]𝐴)))
79	78	pm4.71rd 562	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ∈ (0[,]𝐴) ∧ (𝑝↑𝑘) ≤ 𝐴)))
80		elin 3932	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
81	80	rbaib 538	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ∈ (0[,]𝐴)))
82	81	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ∈ (0[,]𝐴)))
83	82	anbi1d 631	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))))
84	68, 79, 83	3bitr4rd 312	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝↑𝑘) ≤ 𝐴))
85	17, 61	eleqtrdi 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ (ℤ_≥‘1))
86	12	flcld 13766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (⌊‘𝐴) ∈ ℤ)
87		elfz5 13483	. . . . . 6 ⊢ (((𝑝↑𝑘) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → ((𝑝↑𝑘) ∈ (1...(⌊‘𝐴)) ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
88	85, 86, 87	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ∈ (1...(⌊‘𝐴)) ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
89	20, 84, 88	3bitr4d 311	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝↑𝑘) ∈ (1...(⌊‘𝐴))))
90	89	pm5.32da 579	. . 3 ⊢ (𝜑 → (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ (1...(⌊‘𝐴)))))
91	11, 90	bitrid 283	. 2 ⊢ (𝜑 → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ (𝑝↑𝑘) ∈ (1...(⌊‘𝐴)))))
92		fsumvma2.3	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
93		fsumvma2.4	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...(⌊‘𝐴)) ∧ (Λ‘𝑥) = 0)) → 𝐵 = 0)
94	1, 2, 4, 7, 91, 92, 93	fsumvma 27130	1 ⊢ (𝜑 → Σ𝑥 ∈ (1...(⌊‘𝐴))𝐵 = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∩ cin 3915 ⊆ wss 3916 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 Fincfn 8920 ℂcc 11072 ℝcr 11073 0cc0 11074 1c1 11075 · cmul 11079 ≤ cle 11215 / cdiv 11841 ℕcn 12187 2c2 12242 ℕ₀cn0 12448 ℤcz 12535 ℤ_≥cuz 12799 ℝ⁺crp 12957 [,]cicc 13315 ...cfz 13474 ⌊cfl 13758 ↑cexp 14032 Σcsu 15658 ℙcprime 16647 logclog 26469 Λcvma 27008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-dju 9860 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-ioc 13317 df-ico 13318 df-icc 13319 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-fac 14245 df-bc 14274 df-hash 14302 df-shft 15039 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-limsup 15443 df-clim 15460 df-rlim 15461 df-sum 15659 df-ef 16039 df-sin 16041 df-cos 16042 df-pi 16044 df-dvds 16229 df-gcd 16471 df-prm 16648 df-pc 16814 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-xrs 17471 df-qtop 17476 df-imas 17477 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-mulg 19006 df-cntz 19255 df-cmn 19718 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cn 23120 df-cnp 23121 df-haus 23208 df-tx 23455 df-hmeo 23648 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-xms 24214 df-ms 24215 df-tms 24216 df-cncf 24777 df-limc 25773 df-dv 25774 df-log 26471 df-vma 27014

This theorem is referenced by: chpval2 27135 rplogsumlem2 27402 rpvmasumlem 27404

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