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Theorem fsumvma2 27125

Description: Apply fsumvma 27124 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 13938	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
3		fz1ssnn 13516	. . 3 ⊢ (1...(⌊‘𝐴)) ⊆ ℕ
4	3	a1i 11	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆ ℕ)
5		fsumvma2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ppifi 27016	. . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8		elinel2 4165	. . . . 5 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
9		elfznn 13514	. . . . 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 16644	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	13	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ)
15		nnnn0 12449	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
16	15	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ₀)
17	14, 16	nnexpcld 14210	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℕ)
18	17	nnzd 12556	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℤ)
19		flge 13767	. . . . . 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 12993	. . . . . . . . . . 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 12556	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℤ)
26		relogexp 26505	. . . . . . . . . . 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 5117	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
29	24	nnred 12201	. . . . . . . . . 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 11177	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 ∈ ℝ)
32	14	nnred 12201	. . . . . . . . . . . . . 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 12235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝑝)
35		0red 11177	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ∈ ℝ)
36	14	nnnn0d 12503	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ₀)
37	36	nn0ge0d 12506	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ≤ 𝑝)
38		elicc2 13372	. . . . . . . . . . . . . . . . 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 11334	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝐴)
45	30, 44	elrpd 12992	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ⁺)
46	45	relogcld 26532	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝐴) ∈ ℝ)
47		prmuz2 16666	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
48		eluzelre 12804	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℝ)
49		eluz2gt1 12879	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
50	48, 49	rplogcld 26538	. . . . . . . . . . 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 13044	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
53	46, 51	rerpdivcld 13026	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
54		flge 13767	. . . . . . . . . 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 12993	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℝ⁺)
59	58, 45	logled 26536	. . . . . . . 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 12836	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
62	60, 61	eleqtrdi 2838	. . . . . . . . . 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 13760	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
65		elfz5 13477	. . . . . . . . 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 12202	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℂ)
70	69	exp1d 14106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) = 𝑝)
71	14	nnge1d 12234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 1 ≤ 𝑝)
72	32, 71, 62	leexp2ad 14219	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) ≤ (𝑝↑𝑘))
73	70, 72	eqbrtrrd 5131	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ≤ (𝑝↑𝑘))
74	17	nnred 12201	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℝ)
75		letr 11268	. . . . . . . . . 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 3930	. . . . . . . . 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 2838	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ (ℤ_≥‘1))
86	12	flcld 13760	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (⌊‘𝐴) ∈ ℤ)
87		elfz5 13477	. . . . . 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 27124	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 3913 ⊆ wss 3914 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 · cmul 11073 ≤ cle 11209 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 [,]cicc 13309 ...cfz 13468 ⌊cfl 13752 ↑cexp 14026 Σcsu 15652 ℙcprime 16641 logclog 26463 Λcvma 27002

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-vma 27008

This theorem is referenced by: chpval2 27129 rplogsumlem2 27396 rpvmasumlem 27398

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