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Theorem chtdif 16183

Description: The difference of the Chebyshev function at two points sums the logarithms of the primes in an interval. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Assertion**
Ref	Expression
chtdif	⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))

Distinct variable groups: 𝑀,𝑝 𝑁,𝑝

Proof of Theorem chtdif Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluzelz 9940	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
2		zq 10035	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
3	1, 2	syl 14	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℚ)
4		chtqval 16164	. . . . 5 ⊢ (𝑁 ∈ ℚ → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
5	3, 4	syl 14	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (θ‘𝑁) = Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝))
6		eluzel2 9935	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
7		2z 9676	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
8		zmincl 12020	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 2 ∈ ℤ) → inf({𝑀, 2}, ℝ, < ) ∈ ℤ)
9	6, 7, 8	sylancl 417	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → inf({𝑀, 2}, ℝ, < ) ∈ ℤ)
10	7	a1i 9	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 2 ∈ ℤ)
11	6	zred 9772	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℝ)
12		2re 9376	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
13		min2inf 12014	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℝ ∧ 2 ∈ ℝ) → inf({𝑀, 2}, ℝ, < ) ≤ 2)
14	11, 12, 13	sylancl 417	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → inf({𝑀, 2}, ℝ, < ) ≤ 2)
15		eluz2 9936	. . . . . . . . 9 ⊢ (2 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < )) ↔ (inf({𝑀, 2}, ℝ, < ) ∈ ℤ ∧ 2 ∈ ℤ ∧ inf({𝑀, 2}, ℝ, < ) ≤ 2))
16	9, 10, 14, 15	syl3anbrc 1212	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 2 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < )))
17		ppiqsval2 16160	. . . . . . . 8 ⊢ ((𝑁 ∈ ℚ ∧ 2 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < ))) → ((0[,]𝑁) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑁)) ∩ ℙ))
18	3, 16, 17	syl2anc 415	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((0[,]𝑁) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑁)) ∩ ℙ))
19		flid 10732	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
20	1, 19	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (⌊‘𝑁) = 𝑁)
21	20	oveq2d 6101	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (inf({𝑀, 2}, ℝ, < )...(⌊‘𝑁)) = (inf({𝑀, 2}, ℝ, < )...𝑁))
22	21	ineq1d 3431	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑁)) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
23	18, 22	eqtrd 2271	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((0[,]𝑁) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
24	23	sumeq1d 12148	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)(log‘𝑝))
25	11	ltp1d 9262	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 < (𝑀 + 1))
26		fzdisj 10467	. . . . . . . . 9 ⊢ (𝑀 < (𝑀 + 1) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
28	27	ineq1d 3431	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ((𝑀 + 1)...𝑁)) ∩ ℙ) = (∅ ∩ ℙ))
29		inindir 3449	. . . . . . 7 ⊢ (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ((𝑀 + 1)...𝑁)) ∩ ℙ) = (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∩ (((𝑀 + 1)...𝑁) ∩ ℙ))
30		0in 3558	. . . . . . 7 ⊢ (∅ ∩ ℙ) = ∅
31	28, 29, 30	3eqtr3g 2294	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∩ (((𝑀 + 1)...𝑁) ∩ ℙ)) = ∅)
32		min1inf 12013	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℝ ∧ 2 ∈ ℝ) → inf({𝑀, 2}, ℝ, < ) ≤ 𝑀)
33	11, 12, 32	sylancl 417	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → inf({𝑀, 2}, ℝ, < ) ≤ 𝑀)
34		eluz2 9936	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < )) ↔ (inf({𝑀, 2}, ℝ, < ) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ inf({𝑀, 2}, ℝ, < ) ≤ 𝑀))
35	9, 6, 33, 34	syl3anbrc 1212	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < )))
36		id 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (ℤ_≥‘𝑀))
37		elfzuzb 10432	. . . . . . . . . 10 ⊢ (𝑀 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) ↔ (𝑀 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < )) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)))
38	35, 36, 37	sylanbrc 421	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁))
39		fzsplit 10466	. . . . . . . . 9 ⊢ (𝑀 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) → (inf({𝑀, 2}, ℝ, < )...𝑁) = ((inf({𝑀, 2}, ℝ, < )...𝑀) ∪ ((𝑀 + 1)...𝑁)))
40	38, 39	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (inf({𝑀, 2}, ℝ, < )...𝑁) = ((inf({𝑀, 2}, ℝ, < )...𝑀) ∪ ((𝑀 + 1)...𝑁)))
41	40	ineq1d 3431	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) = (((inf({𝑀, 2}, ℝ, < )...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∩ ℙ))
42		indir 3480	. . . . . . 7 ⊢ (((inf({𝑀, 2}, ℝ, < )...𝑀) ∪ ((𝑀 + 1)...𝑁)) ∩ ℙ) = (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∪ (((𝑀 + 1)...𝑁) ∩ ℙ))
43	41, 42	eqtrdi 2287	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) = (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∪ (((𝑀 + 1)...𝑁) ∩ ℙ)))
44	9, 1	fzfigd 10881	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (inf({𝑀, 2}, ℝ, < )...𝑁) ∈ Fin)
45		inss1 3451	. . . . . . . 8 ⊢ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑁)
46	45	a1i 9	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑁))
47		elfzelz 10438	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) → 𝑥 ∈ ℤ)
48	47	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → 𝑥 ∈ ℤ)
49	9	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → inf({𝑀, 2}, ℝ, < ) ∈ ℤ)
50	1	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → 𝑁 ∈ ℤ)
51		fzdcel 10454	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℤ ∧ inf({𝑀, 2}, ℝ, < ) ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁))
52	48, 49, 50, 51	syl3anc 1278	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → DECID 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁))
53		prmdcz 12925	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℤ → DECID 𝑥 ∈ ℙ)
54	47, 53	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) → DECID 𝑥 ∈ ℙ)
55	54	adantl 277	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → DECID 𝑥 ∈ ℙ)
56	52, 55	dcand 945	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → DECID (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) ∧ 𝑥 ∈ ℙ))
57		elin 3412	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ↔ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) ∧ 𝑥 ∈ ℙ))
58	57	dcbii 852	. . . . . . . . 9 ⊢ (DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ↔ DECID (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁) ∧ 𝑥 ∈ ℙ))
59	56, 58	sylibr 134	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)) → DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
60	59	ralrimiva 2623	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ∀𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
61		ssfidc 7245	. . . . . . 7 ⊢ (((inf({𝑀, 2}, ℝ, < )...𝑁) ∈ Fin ∧ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑁) ∧ ∀𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑁)DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ∈ Fin)
62	44, 46, 60, 61	syl3anc 1278	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ) ∈ Fin)
63		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
64	63	elin2d 3419	. . . . . . . . . 10 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → 𝑝 ∈ ℙ)
65		prmnn 12904	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
66	64, 65	syl 14	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → 𝑝 ∈ ℕ)
67	66	nnrpd 10105	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
68	67	relogcld 16034	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
69	68	recnd 8354	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
70	31, 43, 62, 69	fsumsplit 12190	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)(log‘𝑝) = (Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)))
71	24, 70	eqtrd 2271	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = (Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)))
72	5, 71	eqtrd 2271	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (θ‘𝑁) = (Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)))
73		zq 10035	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ)
74	6, 73	syl 14	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℚ)
75		chtqval 16164	. . . . 5 ⊢ (𝑀 ∈ ℚ → (θ‘𝑀) = Σ𝑝 ∈ ((0[,]𝑀) ∩ ℙ)(log‘𝑝))
76	74, 75	syl 14	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (θ‘𝑀) = Σ𝑝 ∈ ((0[,]𝑀) ∩ ℙ)(log‘𝑝))
77		ppiqsval2 16160	. . . . . . 7 ⊢ ((𝑀 ∈ ℚ ∧ 2 ∈ (ℤ_≥‘inf({𝑀, 2}, ℝ, < ))) → ((0[,]𝑀) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑀)) ∩ ℙ))
78	74, 16, 77	syl2anc 415	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((0[,]𝑀) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑀)) ∩ ℙ))
79		flid 10732	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (⌊‘𝑀) = 𝑀)
80	6, 79	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (⌊‘𝑀) = 𝑀)
81	80	oveq2d 6101	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (inf({𝑀, 2}, ℝ, < )...(⌊‘𝑀)) = (inf({𝑀, 2}, ℝ, < )...𝑀))
82	81	ineq1d 3431	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...(⌊‘𝑀)) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ))
83	78, 82	eqtrd 2271	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((0[,]𝑀) ∩ ℙ) = ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ))
84	83	sumeq1d 12148	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((0[,]𝑀) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝))
85	76, 84	eqtrd 2271	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (θ‘𝑀) = Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝))
86	72, 85	oveq12d 6103	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = ((Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)) − Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝)))
87	9, 6	fzfigd 10881	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (inf({𝑀, 2}, ℝ, < )...𝑀) ∈ Fin)
88		inss1 3451	. . . . . 6 ⊢ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑀)
89	88	a1i 9	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑀))
90		elfzelz 10438	. . . . . . . . . 10 ⊢ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀) → 𝑥 ∈ ℤ)
91	90	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → 𝑥 ∈ ℤ)
92	9	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → inf({𝑀, 2}, ℝ, < ) ∈ ℤ)
93	6	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → 𝑀 ∈ ℤ)
94		fzdcel 10454	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ inf({𝑀, 2}, ℝ, < ) ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀))
95	91, 92, 93, 94	syl3anc 1278	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → DECID 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀))
96	90, 53	syl 14	. . . . . . . . 9 ⊢ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀) → DECID 𝑥 ∈ ℙ)
97	96	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → DECID 𝑥 ∈ ℙ)
98	95, 97	dcand 945	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → DECID (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀) ∧ 𝑥 ∈ ℙ))
99		elin 3412	. . . . . . . 8 ⊢ (𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ↔ (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀) ∧ 𝑥 ∈ ℙ))
100	99	dcbii 852	. . . . . . 7 ⊢ (DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ↔ DECID (𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀) ∧ 𝑥 ∈ ℙ))
101	98, 100	sylibr 134	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)) → DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ))
102	101	ralrimiva 2623	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ∀𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ))
103		ssfidc 7245	. . . . 5 ⊢ (((inf({𝑀, 2}, ℝ, < )...𝑀) ∈ Fin ∧ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ⊆ (inf({𝑀, 2}, ℝ, < )...𝑀) ∧ ∀𝑥 ∈ (inf({𝑀, 2}, ℝ, < )...𝑀)DECID 𝑥 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∈ Fin)
104	87, 89, 102, 103	syl3anc 1278	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∈ Fin)
105		ssun1 3392	. . . . . . 7 ⊢ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ⊆ (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∪ (((𝑀 + 1)...𝑁) ∩ ℙ))
106	105, 43	sseqtrrid 3299	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ⊆ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
107	106	sselda 3248	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)) → 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
108	107, 69	syldan 282	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
109	104, 108	fsumcl 12183	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) ∈ ℂ)
110	6	peano2zd 9775	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ ℤ)
111	110, 1	fzfigd 10881	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((𝑀 + 1)...𝑁) ∈ Fin)
112		inss1 3451	. . . . . 6 ⊢ (((𝑀 + 1)...𝑁) ∩ ℙ) ⊆ ((𝑀 + 1)...𝑁)
113	112	a1i 9	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (((𝑀 + 1)...𝑁) ∩ ℙ) ⊆ ((𝑀 + 1)...𝑁))
114		elfzelz 10438	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑀 + 1)...𝑁) → 𝑥 ∈ ℤ)
115	114	adantl 277	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑥 ∈ ℤ)
116	110	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝑀 + 1) ∈ ℤ)
117	1	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → 𝑁 ∈ ℤ)
118		fzdcel 10454	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ (𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ∈ ((𝑀 + 1)...𝑁))
119	115, 116, 117, 118	syl3anc 1278	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → DECID 𝑥 ∈ ((𝑀 + 1)...𝑁))
120	114, 53	syl 14	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝑀 + 1)...𝑁) → DECID 𝑥 ∈ ℙ)
121	120	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → DECID 𝑥 ∈ ℙ)
122	119, 121	dcand 945	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → DECID (𝑥 ∈ ((𝑀 + 1)...𝑁) ∧ 𝑥 ∈ ℙ))
123		elin 3412	. . . . . . . 8 ⊢ (𝑥 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ) ↔ (𝑥 ∈ ((𝑀 + 1)...𝑁) ∧ 𝑥 ∈ ℙ))
124	123	dcbii 852	. . . . . . 7 ⊢ (DECID 𝑥 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ) ↔ DECID (𝑥 ∈ ((𝑀 + 1)...𝑁) ∧ 𝑥 ∈ ℙ))
125	122, 124	sylibr 134	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → DECID 𝑥 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ))
126	125	ralrimiva 2623	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ∀𝑥 ∈ ((𝑀 + 1)...𝑁)DECID 𝑥 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ))
127		ssfidc 7245	. . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ Fin ∧ (((𝑀 + 1)...𝑁) ∩ ℙ) ⊆ ((𝑀 + 1)...𝑁) ∧ ∀𝑥 ∈ ((𝑀 + 1)...𝑁)DECID 𝑥 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)) → (((𝑀 + 1)...𝑁) ∩ ℙ) ∈ Fin)
128	111, 113, 126, 127	syl3anc 1278	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (((𝑀 + 1)...𝑁) ∩ ℙ) ∈ Fin)
129		ssun2 3393	. . . . . . 7 ⊢ (((𝑀 + 1)...𝑁) ∩ ℙ) ⊆ (((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ) ∪ (((𝑀 + 1)...𝑁) ∩ ℙ))
130	129, 43	sseqtrrid 3299	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (((𝑀 + 1)...𝑁) ∩ ℙ) ⊆ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
131	130	sselda 3248	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)) → 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ))
132	131, 69	syldan 282	. . . 4 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
133	128, 132	fsumcl 12183	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝) ∈ ℂ)
134	109, 133	pncan2d 8640	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) + Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝)) − Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))
135	86, 134	eqtrd 2271	1 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → ((θ‘𝑁) − (θ‘𝑀)) = Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∪ cun 3218 ∩ cin 3219 ⊆ wss 3220 ∅c0 3520 {cpr 3710 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 infcinf 7323 ℂcc 8177 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 < clt 8360 ≤ cle 8361 − cmin 8498 ℕcn 9306 2c2 9357 ℤcz 9648 ℤ_≥cuz 9930 ℚcq 10028 [,]cicc 10303 ...cfz 10421 ⌊cfl 10713 Σcsu 12135 ℙcprime 12901 logclog 16007 θccht 16152

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-pre-suploc 8300 ax-addf 8301 ax-mulf 8302

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-n0 9568 df-z 9649 df-uz 9931 df-q 10029 df-rp 10065 df-xneg 10184 df-xadd 10185 df-ioo 10304 df-ico 10306 df-icc 10307 df-fz 10422 df-fzo 10560 df-fl 10715 df-mod 10773 df-seqfrec 10898 df-exp 10989 df-fac 11178 df-bc 11200 df-ihash 11229 df-shft 11594 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136 df-ef 12431 df-e 12432 df-dvds 12571 df-prm 12902 df-rest 13644 df-topgen 13663 df-psmet 14929 df-xmet 14930 df-met 14931 df-bl 14932 df-mopn 14933 df-top 15148 df-topon 15161 df-bases 15193 df-ntr 15246 df-cn 15338 df-cnp 15339 df-tx 15403 df-cncf 15721 df-limced 15806 df-dvap 15807 df-relog 16009 df-cht 16155

This theorem is used by: efchtqdvds 16184

W3C validator