chtdif - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > chtdif		GIF version

Theorem chtdif 16281

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 9946	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
2		zq 10041	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℚ)
3	1, 2	syl 14	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℚ)
4		chtqval 16262	. . . . 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 9941	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
7		2z 9681	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
8		zmincl 12028	. . . . . . . . . 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 9778	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℝ)
12		2re 9383	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
13		min2inf 12022	. . . . . . . . . 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 9942	. . . . . . . . 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 16258	. . . . . . . 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 10739	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (⌊‘𝑁) = 𝑁)
20	1, 19	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (⌊‘𝑁) = 𝑁)
21	20	oveq2d 6104	. . . . . . . 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 12156	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((0[,]𝑁) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)(log‘𝑝))
25	11	ltp1d 9268	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 < (𝑀 + 1))
26		fzdisj 10473	. . . . . . . . 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 12021	. . . . . . . . . . . 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 9942	. . . . . . . . . . 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 10438	. . . . . . . . . 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 10472	. . . . . . . . 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 10888	. . . . . . 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 10444	. . . . . . . . . . . 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 10460	. . . . . . . . . . 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 12933	. . . . . . . . . . . 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 7251	. . . . . . 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 12912	. . . . . . . . . 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 10111	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → 𝑝 ∈ ℝ⁺)
68	67	relogcld 16129	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → (log‘𝑝) ∈ ℝ)
69	68	recnd 8360	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ_≥‘𝑀) ∧ 𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑁) ∩ ℙ)) → (log‘𝑝) ∈ ℂ)
70	31, 43, 62, 69	fsumsplit 12198	. . . . 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 10041	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℚ)
74	6, 73	syl 14	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℚ)
75		chtqval 16262	. . . . 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 16258	. . . . . . 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 10739	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (⌊‘𝑀) = 𝑀)
80	6, 79	syl 14	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (⌊‘𝑀) = 𝑀)
81	80	oveq2d 6104	. . . . . . 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 12156	. . . 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 6106	. 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 10888	. . . . 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 10444	. . . . . . . . . 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 10460	. . . . . . . . 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 7251	. . . . 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 12191	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ ((inf({𝑀, 2}, ℝ, < )...𝑀) ∩ ℙ)(log‘𝑝) ∈ ℂ)
110	6	peano2zd 9781	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑀 + 1) ∈ ℤ)
111	110, 1	fzfigd 10888	. . . . 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 10444	. . . . . . . . . 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 10460	. . . . . . . . 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 7251	. . . . 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 12191	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → Σ𝑝 ∈ (((𝑀 + 1)...𝑁) ∩ ℙ)(log‘𝑝) ∈ ℂ)
134	109, 133	pncan2d 8646	. 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 4131 ‘cfv 5380 (class class class)co 6088 Fincfn 7028 infcinf 7329 ℂcc 8183 ℝcr 8184 0cc0 8185 1c1 8186 + caddc 8188 < clt 8366 ≤ cle 8367 − cmin 8504 ℕcn 9313 2c2 9364 ℤcz 9653 ℤ_≥cuz 9936 ℚcq 10034 [,]cicc 10309 ...cfz 10427 ⌊cfl 10719 Σcsu 12143 ℙcprime 12909 logclog 16102 θccht 16250

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305 ax-pre-suploc 8306 ax-addf 8307 ax-mulf 8308

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6581 df-irdg 6646 df-frec 6667 df-1o 6692 df-2o 6693 df-oadd 6696 df-er 6812 df-map 6929 df-pm 6930 df-en 7029 df-dom 7030 df-fin 7031 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9654 df-uz 9937 df-q 10035 df-rp 10071 df-xneg 10190 df-xadd 10191 df-ioo 10310 df-ico 10312 df-icc 10313 df-fz 10428 df-fzo 10566 df-fl 10721 df-mod 10780 df-seqfrec 10905 df-exp 10996 df-fac 11185 df-bc 11207 df-ihash 11236 df-shft 11601 df-cj 11628 df-re 11629 df-im 11630 df-rsqrt 11785 df-abs 11786 df-clim 12069 df-sumdc 12144 df-ef 12439 df-e 12440 df-dvds 12579 df-prm 12910 df-rest 13653 df-topgen 13672 df-psmet 14988 df-xmet 14989 df-met 14990 df-bl 14991 df-mopn 14992 df-top 15241 df-topon 15254 df-bases 15286 df-ntr 15339 df-cn 15431 df-cnp 15432 df-tx 15496 df-cncf 15814 df-limced 15899 df-dvap 15900 df-relog 16104 df-cht 16253

This theorem is used by: efchtqdvds 16282

W3C validator