Theorem chtdif 16276

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 e. ( ZZ>= ` M ) -> ( ( theta ` N ) - ( theta ` M ) ) = sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( 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 9948	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
2		zq 10043	. . . . . 6 |- ( N e. ZZ -> N e. QQ )
3	1, 2	syl 14	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. QQ )
4		chtqval 16257	. . . . 5 |- ( N e. QQ -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) )
5	3, 4	syl 14	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) )
6		eluzel2 9943	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
7		2z 9684	. . . . . . . . . 10 |- 2 e. ZZ
8		zmincl 12030	. . . . . . . . . 10 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
9	6, 7, 8	sylancl 417	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
10	7	a1i 9	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> 2 e. ZZ )
11	6	zred 9780	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. RR )
12		2re 9384	. . . . . . . . . 10 |- 2 e. RR
13		min2inf 12024	. . . . . . . . . 10 |- ( ( M e. RR /\ 2 e. RR ) -> inf ( { M , 2 } , RR , < ) <_ 2 )
14	11, 12, 13	sylancl 417	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) <_ 2 )
15		eluz2 9944	. . . . . . . . 9 |- ( 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) <-> (inf ( { M , 2 } , RR , < ) e. ZZ /\ 2 e. ZZ /\ inf ( { M , 2 } , RR , < ) <_ 2 ) )
16	9, 10, 14, 15	syl3anbrc 1212	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) )
17		ppiqsval2 16253	. . . . . . . 8 |- ( ( N e. QQ /\ 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) ) -> ( ( 0 [,] N ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` N ) ) i^i Prime ) )
18	3, 16, 17	syl2anc 415	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ( 0 [,] N ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` N ) ) i^i Prime ) )
19		flid 10741	. . . . . . . . . 10 |- ( N e. ZZ -> ( |_ ` N ) = N )
20	1, 19	syl 14	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> ( |_ ` N ) = N )
21	20	oveq2d 6104	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... ( |_ ` N ) ) = (inf ( { M , 2 } , RR , < ) ... N ) )
22	21	ineq1d 3431	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` N ) ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
23	18, 22	eqtrd 2271	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( 0 [,] N ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
24	23	sumeq1d 12158	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) = sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ( log ` p ) )
25	11	ltp1d 9269	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M < ( M + 1 ) )
26		fzdisj 10475	. . . . . . . . 9 |- ( M < ( M + 1 ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
27	25, 26	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
28	27	ineq1d 3431	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( (/) i^i Prime ) )
29		inindir 3449	. . . . . . 7 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) )
30		0in 3558	. . . . . . 7 |- ( (/) i^i Prime ) = (/)
31	28, 29, 30	3eqtr3g 2294	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) )
32		min1inf 12023	. . . . . . . . . . . 12 |- ( ( M e. RR /\ 2 e. RR ) -> inf ( { M , 2 } , RR , < ) <_ M )
33	11, 12, 32	sylancl 417	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) <_ M )
34		eluz2 9944	. . . . . . . . . . 11 |- ( M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) <-> (inf ( { M , 2 } , RR , < ) e. ZZ /\ M e. ZZ /\ inf ( { M , 2 } , RR , < ) <_ M ) )
35	9, 6, 33, 34	syl3anbrc 1212	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) )
36		id 19	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> N e. ( ZZ>= ` M ) )
37		elfzuzb 10440	. . . . . . . . . 10 |- ( M e. (inf ( { M , 2 } , RR , < ) ... N ) <-> ( M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) /\ N e. ( ZZ>= ` M ) ) )
38	35, 36, 37	sylanbrc 421	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. (inf ( { M , 2 } , RR , < ) ... N ) )
39		fzsplit 10474	. . . . . . . . 9 |- ( M e. (inf ( { M , 2 } , RR , < ) ... N ) -> (inf ( { M , 2 } , RR , < ) ... N ) = ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) )
40	38, 39	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... N ) = ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) )
41	40	ineq1d 3431	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) )
42		indir 3480	. . . . . . 7 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) )
43	41, 42	eqtrdi 2287	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) )
44	9, 1	fzfigd 10890	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... N ) e. Fin )
45		inss1 3451	. . . . . . . 8 |- ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... N )
46	45	a1i 9	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... N ) )
47		elfzelz 10446	. . . . . . . . . . . 12 |- ( x e. (inf ( { M , 2 } , RR , < ) ... N ) -> x e. ZZ )
48	47	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> x e. ZZ )
49	9	adantr 276	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
50	1	adantr 276	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> N e. ZZ )
51		fzdcel 10462	. . . . . . . . . . 11 |- ( ( x e. ZZ /\ inf ( { M , 2 } , RR , < ) e. ZZ /\ N e. ZZ ) -> DECID x e. (inf ( { M , 2 } , RR , < ) ... N ) )
52	48, 49, 50, 51	syl3anc 1278	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> DECID x e. (inf ( { M , 2 } , RR , < ) ... N ) )
53		prmdcz 12935	. . . . . . . . . . . 12 |- ( x e. ZZ -> DECID x e. Prime )
54	47, 53	syl 14	. . . . . . . . . . 11 |- ( x e. (inf ( { M , 2 } , RR , < ) ... N ) -> DECID x e. Prime )
55	54	adantl 277	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> DECID x e. Prime )
56	52, 55	dcand 945	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... N ) /\ x e. Prime ) )
57		elin 3412	. . . . . . . . . 10 |- ( x e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) <-> ( x e. (inf ( { M , 2 } , RR , < ) ... N ) /\ x e. Prime ) )
58	57	dcbii 852	. . . . . . . . 9 |- (DECID x e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) <-> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... N ) /\ x e. Prime ) )
59	56, 58	sylibr 134	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... N ) ) -> DECID x e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
60	59	ralrimiva 2623	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> A. x e. (inf ( { M , 2 } , RR , < ) ... N )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
61		ssfidc 7250	. . . . . . 7 |- ( ( (inf ( { M , 2 } , RR , < ) ... N ) e. Fin /\ ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... N ) /\ A. x e. (inf ( { M , 2 } , RR , < ) ... N )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) e. Fin )
62	44, 46, 60, 61	syl3anc 1278	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) e. Fin )
63		simpr 110	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
64	63	elin2d 3419	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> p e. Prime )
65		prmnn 12914	. . . . . . . . . 10 |- ( p e. Prime -> p e. NN )
66	64, 65	syl 14	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> p e. NN )
67	66	nnrpd 10113	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> p e. RR+ )
68	67	relogcld 16124	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> ( log ` p ) e. RR )
69	68	recnd 8361	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) -> ( log ` p ) e. CC )
70	31, 43, 62, 69	fsumsplit 12200	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ( log ` p ) = ( sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) + sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) ) )
71	24, 70	eqtrd 2271	. . . 4 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) = ( sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) + sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) ) )
72	5, 71	eqtrd 2271	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( theta ` N ) = ( sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) + sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) ) )
73		zq 10043	. . . . . 6 |- ( M e. ZZ -> M e. QQ )
74	6, 73	syl 14	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> M e. QQ )
75		chtqval 16257	. . . . 5 |- ( M e. QQ -> ( theta ` M ) = sum_ p e. ( ( 0 [,] M ) i^i Prime ) ( log ` p ) )
76	74, 75	syl 14	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( theta ` M ) = sum_ p e. ( ( 0 [,] M ) i^i Prime ) ( log ` p ) )
77		ppiqsval2 16253	. . . . . . 7 |- ( ( M e. QQ /\ 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) ) -> ( ( 0 [,] M ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` M ) ) i^i Prime ) )
78	74, 16, 77	syl2anc 415	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( 0 [,] M ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` M ) ) i^i Prime ) )
79		flid 10741	. . . . . . . . 9 |- ( M e. ZZ -> ( |_ ` M ) = M )
80	6, 79	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> ( |_ ` M ) = M )
81	80	oveq2d 6104	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... ( |_ ` M ) ) = (inf ( { M , 2 } , RR , < ) ... M ) )
82	81	ineq1d 3431	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... ( |_ ` M ) ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
83	78, 82	eqtrd 2271	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( 0 [,] M ) i^i Prime ) = ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
84	83	sumeq1d 12158	. . . 4 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( ( 0 [,] M ) i^i Prime ) ( log ` p ) = sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) )
85	76, 84	eqtrd 2271	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( theta ` M ) = sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) )
86	72, 85	oveq12d 6106	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( theta ` N ) - ( theta ` M ) ) = ( ( sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) + sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) ) - sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) ) )
87	9, 6	fzfigd 10890	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... M ) e. Fin )
88		inss1 3451	. . . . . 6 |- ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M )
89	88	a1i 9	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M ) )
90		elfzelz 10446	. . . . . . . . . 10 |- ( x e. (inf ( { M , 2 } , RR , < ) ... M ) -> x e. ZZ )
91	90	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> x e. ZZ )
92	9	adantr 276	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
93	6	adantr 276	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> M e. ZZ )
94		fzdcel 10462	. . . . . . . . 9 |- ( ( x e. ZZ /\ inf ( { M , 2 } , RR , < ) e. ZZ /\ M e. ZZ ) -> DECID x e. (inf ( { M , 2 } , RR , < ) ... M ) )
95	91, 92, 93, 94	syl3anc 1278	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. (inf ( { M , 2 } , RR , < ) ... M ) )
96	90, 53	syl 14	. . . . . . . . 9 |- ( x e. (inf ( { M , 2 } , RR , < ) ... M ) -> DECID x e. Prime )
97	96	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. Prime )
98	95, 97	dcand 945	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
99		elin 3412	. . . . . . . 8 |- ( x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) <-> ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
100	99	dcbii 852	. . . . . . 7 |- (DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) <-> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
101	98, 100	sylibr 134	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
102	101	ralrimiva 2623	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> A. x e. (inf ( { M , 2 } , RR , < ) ... M )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
103		ssfidc 7250	. . . . 5 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) e. Fin /\ ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M ) /\ A. x e. (inf ( { M , 2 } , RR , < ) ... M )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin )
104	87, 89, 102, 103	syl3anc 1278	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin )
105		ssun1 3392	. . . . . . 7 |- ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) )
106	105, 43	sseqtrrid 3299	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
107	106	sselda 3248	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) -> p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
108	107, 69	syldan 282	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) -> ( log ` p ) e. CC )
109	104, 108	fsumcl 12193	. . 3 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) e. CC )
110	6	peano2zd 9783	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( M + 1 ) e. ZZ )
111	110, 1	fzfigd 10890	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( M + 1 ) ... N ) e. Fin )
112		inss1 3451	. . . . . 6 |- ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N )
113	112	a1i 9	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N ) )
114		elfzelz 10446	. . . . . . . . . 10 |- ( x e. ( ( M + 1 ) ... N ) -> x e. ZZ )
115	114	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> x e. ZZ )
116	110	adantr 276	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. ZZ )
117	1	adantr 276	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> N e. ZZ )
118		fzdcel 10462	. . . . . . . . 9 |- ( ( x e. ZZ /\ ( M + 1 ) e. ZZ /\ N e. ZZ ) -> DECID x e. ( ( M + 1 ) ... N ) )
119	115, 116, 117, 118	syl3anc 1278	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. ( ( M + 1 ) ... N ) )
120	114, 53	syl 14	. . . . . . . . 9 |- ( x e. ( ( M + 1 ) ... N ) -> DECID x e. Prime )
121	120	adantl 277	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. Prime )
122	119, 121	dcand 945	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
123		elin 3412	. . . . . . . 8 |- ( x e. ( ( ( M + 1 ) ... N ) i^i Prime ) <-> ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
124	123	dcbii 852	. . . . . . 7 |- (DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) <-> DECID ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
125	122, 124	sylibr 134	. . . . . 6 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) )
126	125	ralrimiva 2623	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> A. x e. ( ( M + 1 ) ... N )DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) )
127		ssfidc 7250	. . . . 5 |- ( ( ( ( M + 1 ) ... N ) e. Fin /\ ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N ) /\ A. x e. ( ( M + 1 ) ... N )DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin )
128	111, 113, 126, 127	syl3anc 1278	. . . 4 |- ( N e. ( ZZ>= ` M ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin )
129		ssun2 3393	. . . . . . 7 |- ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) )
130	129, 43	sseqtrrid 3299	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
131	130	sselda 3248	. . . . 5 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ) -> p e. ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) )
132	131, 69	syldan 282	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ) -> ( log ` p ) e. CC )
133	128, 132	fsumcl 12193	. . 3 |- ( N e. ( ZZ>= ` M ) -> sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) e. CC )
134	109, 133	pncan2d 8647	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) + sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) ) - sum_ p e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ( log ` p ) ) = sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) )
135	86, 134	eqtrd 2271	1 |- ( N e. ( ZZ>= ` M ) -> ( ( theta ` N ) - ( theta ` M ) ) = sum_ p e. ( ( ( M + 1 ) ... N ) i^i Prime ) ( log ` p ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {cpr 3710   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   Fincfn 7027  infcinf 7330   CCcc 8184   RRcr 8185   0cc0 8186   1c1 8187   + caddc 8189   < clt 8367   <_ cle 8368   - cmin 8505   NNcn 9314   2c2 9365   ZZcz 9656   ZZ>=cuz 9938   QQcq 10036   [,]cicc 10311   ...cfz 10429   |_cfl 10721   sum_csu 12145   Primecprime 12911   logclog 16097   thetaccht 16245

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 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-pre-suploc 8307  ax-addf 8308  ax-mulf 8309

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 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-ioo 10312  df-ico 10314  df-icc 10315  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-shft 11603  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-ef 12441  df-e 12442  df-dvds 12581  df-prm 12912  df-rest 13655  df-topgen 13674  df-psmet 14990  df-xmet 14991  df-met 14992  df-bl 14993  df-mopn 14994  df-top 15236  df-topon 15249  df-bases 15281  df-ntr 15334  df-cn 15426  df-cnp 15427  df-tx 15491  df-cncf 15809  df-limced 15894  df-dvap 15895  df-relog 16099  df-cht 16248

This theorem is used by:  efchtqdvds  16277