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Theorem fsumvma2 27158

Description: Apply fsumvma 27157 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 13914	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
3		fz1ssnn 13492	. . 3 ⊢ (1...(⌊‘𝐴)) ⊆ ℕ
4	3	a1i 11	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆ ℕ)
5		fsumvma2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ppifi 27049	. . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8		elinel2 4161	. . . . 5 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
9		elfznn 13490	. . . . 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 16620	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	13	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ)
15		nnnn0 12425	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
16	15	ad2antll 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ₀)
17	14, 16	nnexpcld 14186	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℕ)
18	17	nnzd 12532	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℤ)
19		flge 13743	. . . . . 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 12969	. . . . . . . . . . 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 12532	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℤ)
26		relogexp 26538	. . . . . . . . . . 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 5112	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
29	24	nnred 12177	. . . . . . . . . 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 11153	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 ∈ ℝ)
32	14	nnred 12177	. . . . . . . . . . . . . 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 12211	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝑝)
35		0red 11153	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ∈ ℝ)
36	14	nnnn0d 12479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ₀)
37	36	nn0ge0d 12482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ≤ 𝑝)
38		elicc2 13348	. . . . . . . . . . . . . . . . 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 11310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝐴)
45	30, 44	elrpd 12968	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ⁺)
46	45	relogcld 26565	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝐴) ∈ ℝ)
47		prmuz2 16642	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
48		eluzelre 12780	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℝ)
49		eluz2gt1 12855	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
50	48, 49	rplogcld 26571	. . . . . . . . . . 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 13020	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
53	46, 51	rerpdivcld 13002	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
54		flge 13743	. . . . . . . . . 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 12969	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℝ⁺)
59	58, 45	logled 26569	. . . . . . . 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 12812	. . . . . . . . . . 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 13736	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
65		elfz5 13453	. . . . . . . . 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 12178	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℂ)
70	69	exp1d 14082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) = 𝑝)
71	14	nnge1d 12210	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 1 ≤ 𝑝)
72	32, 71, 62	leexp2ad 14195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) ≤ (𝑝↑𝑘))
73	70, 72	eqbrtrrd 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ≤ (𝑝↑𝑘))
74	17	nnred 12177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℝ)
75		letr 11244	. . . . . . . . . 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 3927	. . . . . . . . 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 13736	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (⌊‘𝐴) ∈ ℤ)
87		elfz5 13453	. . . . . 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 27157	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 3910 ⊆ wss 3911 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 Fincfn 8895 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 ≤ cle 11185 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 [,]cicc 13285 ...cfz 13444 ⌊cfl 13728 ↑cexp 14002 Σcsu 15628 ℙcprime 16617 logclog 26496 Λcvma 27035

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801 df-log 26498 df-vma 27041

This theorem is referenced by: chpval2 27162 rplogsumlem2 27429 rpvmasumlem 27431

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