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Theorem fsumvma2 27165

Description: Apply fsumvma 27164 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 13924	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
3		fz1ssnn 13498	. . 3 ⊢ (1...(⌊‘𝐴)) ⊆ ℕ
4	3	a1i 11	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆ ℕ)
5		fsumvma2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ppifi 27057	. . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8		elinel2 4133	. . . . 5 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
9		elfznn 13496	. . . . 5 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
10	8, 9	anim12i 614	. . . 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 16632	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	13	ad2antrl 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ)
15		nnnn0 12433	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
16	15	ad2antll 730	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ₀)
17	14, 16	nnexpcld 14196	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℕ)
18	17	nnzd 12539	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℤ)
19		flge 13753	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℤ) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
20	12, 18, 19	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
21		simplrl 777	. . . . . . . . . . . . 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 12973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℝ⁺)
24		simplrr 778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℕ)
25	24	nnzd 12539	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℤ)
26		relogexp 26548	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
27	23, 25, 26	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
28	27	breq1d 5084	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
29	24	nnred 12178	. . . . . . . . . 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 11136	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 ∈ ℝ)
32	14	nnred 12178	. . . . . . . . . . . . . 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 12215	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝑝)
35		0red 11136	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ∈ ℝ)
36	14	nnnn0d 12487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ₀)
37	36	nn0ge0d 12490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ≤ 𝑝)
38		elicc2 13353	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
39		df-3an 1089	. . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . 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 11295	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝐴)
45	30, 44	elrpd 12972	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ⁺)
46	45	relogcld 26575	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝐴) ∈ ℝ)
47		prmuz2 16654	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
48		eluzelre 12788	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℝ)
49		eluz2gt1 12859	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
50	48, 49	rplogcld 26581	. . . . . . . . . . 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 13024	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
53	46, 51	rerpdivcld 13006	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
54		flge 13753	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
55	53, 25, 54	syl2anc 585	. . . . . . . . 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 12973	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℝ⁺)
59	58, 45	logled 26579	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
60		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
61		nnuz 12816	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
62	60, 61	eleqtrdi 2845	. . . . . . . . . 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 13746	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
65		elfz5 13459	. . . . . . . . 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 585	. . . . . . . 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 12179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℂ)
70	69	exp1d 14092	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) = 𝑝)
71	14	nnge1d 12214	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 1 ≤ 𝑝)
72	32, 71, 62	leexp2ad 14205	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) ≤ (𝑝↑𝑘))
73	70, 72	eqbrtrrd 5098	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ≤ (𝑝↑𝑘))
74	17	nnred 12178	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℝ)
75		letr 11229	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
76	32, 74, 12, 75	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
77	73, 76	mpand 696	. . . . . . . 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 3901	. . . . . . . . 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 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ∈ (0[,]𝐴)))
83	82	anbi1d 632	. . . . . 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 2845	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ (ℤ_≥‘1))
86	12	flcld 13746	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (⌊‘𝐴) ∈ ℤ)
87		elfz5 13459	. . . . . 6 ⊢ (((𝑝↑𝑘) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → ((𝑝↑𝑘) ∈ (1...(⌊‘𝐴)) ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
88	85, 86, 87	syl2anc 585	. . . . 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 27164	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 1087 = wceq 1542 ∈ wcel 2114 ∩ cin 3884 ⊆ wss 3885 class class class wbr 5074 ‘cfv 6487 (class class class)co 7356 Fincfn 8882 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 · cmul 11032 ≤ cle 11169 / cdiv 11796 ℕcn 12163 2c2 12225 ℕ₀cn0 12426 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 [,]cicc 13290 ...cfz 13450 ⌊cfl 13738 ↑cexp 14012 Σcsu 15637 ℙcprime 16629 logclog 26506 Λcvma 27043

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-dvds 16211 df-gcd 16453 df-prm 16630 df-pc 16797 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-vma 27049

This theorem is referenced by: chpval2 27169 rplogsumlem2 27436 rpvmasumlem 27438

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