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Theorem fsumvma2 27184

Description: Apply fsumvma 27183 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 13971	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
3		fz1ssnn 13564	. . 3 ⊢ (1...(⌊‘𝐴)) ⊆ ℕ
4	3	a1i 11	. 2 ⊢ (𝜑 → (1...(⌊‘𝐴)) ⊆ ℕ)
5		fsumvma2.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
6		ppifi 27075	. . 3 ⊢ (𝐴 ∈ ℝ → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ((0[,]𝐴) ∩ ℙ) ∈ Fin)
8		elinel2 4195	. . . . 5 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ)
9		elfznn 13562	. . . . 5 ⊢ (𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ)
10	8, 9	anim12i 611	. . . 4 ⊢ ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))
11	10	pm4.71ri 559	. . 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 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝐴 ∈ ℝ)
13		prmnn 16645	. . . . . . . . 9 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
14	13	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ)
15		nnnn0 12509	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
16	15	ad2antll 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ₀)
17	14, 16	nnexpcld 14240	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℕ)
18	17	nnzd 12615	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℤ)
19		flge 13803	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℤ) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
20	12, 18, 19	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
21		simplrl 775	. . . . . . . . . . . . 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 13046	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℝ⁺)
24		simplrr 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℕ)
25	24	nnzd 12615	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℤ)
26		relogexp 26567	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
27	23, 25, 26	syl2anc 582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝)))
28	27	breq1d 5158	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴)))
29	24	nnred 12257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ ℝ)
30	12	adantr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ)
31		0red 11247	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 ∈ ℝ)
32	14	nnred 12257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℝ)
33	32	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ∈ ℝ)
34	22	nngt0d 12291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝑝)
35		0red 11247	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ∈ ℝ)
36	14	nnnn0d 12562	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℕ₀)
37	36	nn0ge0d 12565	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 0 ≤ 𝑝)
38		elicc2 13421	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)))
39		df-3an 1086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴))
40	38, 39	bitrdi 286	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑝 ∈ (0[,]𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)))
41	40	baibd 538	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
42	35, 12, 32, 37, 41	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴))
43	42	biimpa 475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑝 ≤ 𝐴)
44	31, 33, 30, 34, 43	ltletrd 11404	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 0 < 𝐴)
45	30, 44	elrpd 13045	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝐴 ∈ ℝ⁺)
46	45	relogcld 26594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (log‘𝐴) ∈ ℝ)
47		prmuz2 16667	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
48		eluzelre 12863	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 𝑝 ∈ ℝ)
49		eluz2gt1 12934	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
50	48, 49	rplogcld 26600	. . . . . . . . . . 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 13097	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝))))
53	46, 51	rerpdivcld 13079	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘𝐴) / (log‘𝑝)) ∈ ℝ)
54		flge 13803	. . . . . . . . . 10 ⊢ ((((log‘𝐴) / (log‘𝑝)) ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
55	53, 25, 54	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
56	28, 52, 55	3bitrd 304	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝)))))
57	17	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℕ)
58	57	nnrpd 13046	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (𝑝↑𝑘) ∈ ℝ⁺)
59	58, 45	logled 26598	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴)))
60		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
61		nnuz 12895	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
62	60, 61	eleqtrdi 2835	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ (ℤ_≥‘1))
63	62	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → 𝑘 ∈ (ℤ_≥‘1))
64	53	flcld 13796	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ)
65		elfz5 13525	. . . . . . . . 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 582	. . . . . . . 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 310	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) ∧ 𝑝 ∈ (0[,]𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))))
68	67	pm5.32da 577	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ (0[,]𝐴) ∧ (𝑝↑𝑘) ≤ 𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))))
69	14	nncnd 12258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ∈ ℂ)
70	69	exp1d 14138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) = 𝑝)
71	14	nnge1d 12290	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 1 ≤ 𝑝)
72	32, 71, 62	leexp2ad 14249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑1) ≤ (𝑝↑𝑘))
73	70, 72	eqbrtrrd 5172	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → 𝑝 ≤ (𝑝↑𝑘))
74	17	nnred 12257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ ℝ)
75		letr 11338	. . . . . . . . . 10 ⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
76	32, 74, 12, 75	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴))
77	73, 76	mpand 693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴))
78	77, 42	sylibrd 258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ∈ (0[,]𝐴)))
79	78	pm4.71rd 561	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ∈ (0[,]𝐴) ∧ (𝑝↑𝑘) ≤ 𝐴)))
80		elin 3961	. . . . . . . . 9 ⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))
81	80	rbaib 537	. . . . . . . 8 ⊢ (𝑝 ∈ ℙ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ∈ (0[,]𝐴)))
82	81	ad2antrl 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ∈ (0[,]𝐴)))
83	82	anbi1d 629	. . . . . 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 311	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝↑𝑘) ≤ 𝐴))
85	17, 61	eleqtrdi 2835	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (𝑝↑𝑘) ∈ (ℤ_≥‘1))
86	12	flcld 13796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → (⌊‘𝐴) ∈ ℤ)
87		elfz5 13525	. . . . . 6 ⊢ (((𝑝↑𝑘) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → ((𝑝↑𝑘) ∈ (1...(⌊‘𝐴)) ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
88	85, 86, 87	syl2anc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝↑𝑘) ∈ (1...(⌊‘𝐴)) ↔ (𝑝↑𝑘) ≤ (⌊‘𝐴)))
89	20, 84, 88	3bitr4d 310	. . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝↑𝑘) ∈ (1...(⌊‘𝐴))))
90	89	pm5.32da 577	. . 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 282	. 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 27183	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3944 ⊆ wss 3945 class class class wbr 5148 ‘cfv 6547 (class class class)co 7417 Fincfn 8962 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 · cmul 11143 ≤ cle 11279 / cdiv 11901 ℕcn 12242 2c2 12297 ℕ₀cn0 12502 ℤcz 12588 ℤ_≥cuz 12852 ℝ⁺crp 13006 [,]cicc 13359 ...cfz 13516 ⌊cfl 13788 ↑cexp 14059 Σcsu 15665 ℙcprime 16642 logclog 26525 Λcvma 27061

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 df-sin 16046 df-cos 16047 df-pi 16049 df-dvds 16232 df-gcd 16470 df-prm 16643 df-pc 16806 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-mulg 19029 df-cntz 19276 df-cmn 19745 df-psmet 21282 df-xmet 21283 df-met 21284 df-bl 21285 df-mopn 21286 df-fbas 21287 df-fg 21288 df-cnfld 21291 df-top 22833 df-topon 22850 df-topsp 22872 df-bases 22886 df-cld 22960 df-ntr 22961 df-cls 22962 df-nei 23039 df-lp 23077 df-perf 23078 df-cn 23168 df-cnp 23169 df-haus 23256 df-tx 23503 df-hmeo 23696 df-fil 23787 df-fm 23879 df-flim 23880 df-flf 23881 df-xms 24263 df-ms 24264 df-tms 24265 df-cncf 24835 df-limc 25832 df-dv 25833 df-log 26527 df-vma 27067

This theorem is referenced by: chpval2 27188 rplogsumlem2 27455 rpvmasumlem 27457

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