Theorem rplogsum 25216

Description: The sum of log p / p over the primes p == A (mod N) is asymptotic to log x / phi ( x ) + O(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 16-Apr-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	|- Z = (ℤ/nℤ ` N )
rpvmasum.l	|- L = ( ZRHom ` Z )
rpvmasum.a	|- ( ph -> N e. NN )
rpvmasum.u	|- U = (Unit ` Z )
rpvmasum.b	|- ( ph -> A e. U )
rpvmasum.t	|- T = ( `' L " { A } )

Assertion

Ref	Expression
rplogsum	|- ( ph -> ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) e. O(1) )

Distinct variable groups:   x, p, A   N, p, x   ph, p, x   T, p, x   U, p, x   Z, p, x   L, p, x

Proof of Theorem rplogsum

Step	Hyp	Ref	Expression
1		rpvmasum.z	. . 3 |- Z = (ℤ/nℤ ` N )
2		rpvmasum.l	. . 3 |- L = ( ZRHom ` Z )
3		rpvmasum.a	. . 3 |- ( ph -> N e. NN )
4		rpvmasum.u	. . 3 |- U = (Unit ` Z )
5		rpvmasum.b	. . 3 |- ( ph -> A e. U )
6		rpvmasum.t	. . 3 |- T = ( `' L " { A } )
7	1, 2, 3, 4, 5, 6	rpvmasum 25215	. 2 |- ( ph -> ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) ) e. O(1) )
8	3	phicld 15477	. . . . . . 7 |- ( ph -> ( phi ` N ) e. NN )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( phi ` N ) e. NN )
10	9	nncnd 11036	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( phi ` N ) e. CC )
11		fzfid 12772	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin )
12		inss1 3833	. . . . . . . 8 |- ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) )
13		ssfi 8180	. . . . . . . 8 |- ( ( ( 1 ... ( |_ ` x ) ) e. Fin /\ ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) e. Fin )
14	11, 12, 13	sylancl 694	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) e. Fin )
15		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) )
16	12, 15	sseldi 3601	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> p e. ( 1 ... ( |_ ` x ) ) )
17		elfznn 12370	. . . . . . . . 9 |- ( p e. ( 1 ... ( |_ ` x ) ) -> p e. NN )
18	16, 17	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> p e. NN )
19		vmacl 24844	. . . . . . . . 9 |- ( p e. NN -> (Λ ` p ) e. RR )
20		nndivre 11056	. . . . . . . . 9 |- ( ( (Λ ` p ) e. RR /\ p e. NN ) -> ( (Λ ` p ) / p ) e. RR )
21	19, 20	mpancom 703	. . . . . . . 8 |- ( p e. NN -> ( (Λ ` p ) / p ) e. RR )
22	18, 21	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( (Λ ` p ) / p ) e. RR )
23	14, 22	fsumrecl 14465	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) e. RR )
24	23	recnd 10068	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) e. CC )
25	10, 24	mulcld 10060	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) e. CC )
26		relogcl 24322	. . . . . 6 |- ( x e. RR+ -> ( log ` x ) e. RR )
27	26	adantl 482	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. RR )
28	27	recnd 10068	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. CC )
29	25, 28	subcld 10392	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) e. CC )
30		inss1 3833	. . . . . . . 8 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( 1 ... ( |_ ` x ) )
31		ssfi 8180	. . . . . . . 8 |- ( ( ( 1 ... ( |_ ` x ) ) e. Fin /\ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( 1 ... ( |_ ` x ) ) ) -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) e. Fin )
32	11, 30, 31	sylancl 694	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) e. Fin )
33		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) )
34	30, 33	sseldi 3601	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> p e. ( 1 ... ( |_ ` x ) ) )
35	34, 17	syl 17	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> p e. NN )
36		nnrp 11842	. . . . . . . . . 10 |- ( p e. NN -> p e. RR+ )
37	36	relogcld 24369	. . . . . . . . 9 |- ( p e. NN -> ( log ` p ) e. RR )
38	37, 36	rerpdivcld 11903	. . . . . . . 8 |- ( p e. NN -> ( ( log ` p ) / p ) e. RR )
39	35, 38	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> ( ( log ` p ) / p ) e. RR )
40	32, 39	fsumrecl 14465	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) e. RR )
41	40	recnd 10068	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) e. CC )
42	10, 41	mulcld 10060	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) e. CC )
43	42, 28	subcld 10392	. . 3 |- ( ( ph /\ x e. RR+ ) -> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) e. CC )
44	10, 24, 41	subdid 10486	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) ) = ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) ) )
45	19	recnd 10068	. . . . . . . . . . 11 |- ( p e. NN -> (Λ ` p ) e. CC )
46		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
47		ifcl 4130	. . . . . . . . . . . . 13 |- ( ( ( log ` p ) e. RR /\ 0 e. RR ) -> if ( p e. Prime , ( log ` p ) , 0 ) e. RR )
48	37, 46, 47	sylancl 694	. . . . . . . . . . . 12 |- ( p e. NN -> if ( p e. Prime , ( log ` p ) , 0 ) e. RR )
49	48	recnd 10068	. . . . . . . . . . 11 |- ( p e. NN -> if ( p e. Prime , ( log ` p ) , 0 ) e. CC )
50	36	rpcnne0d 11881	. . . . . . . . . . 11 |- ( p e. NN -> ( p e. CC /\ p =/= 0 ) )
51		divsubdir 10721	. . . . . . . . . . 11 |- ( ( (Λ ` p ) e. CC /\ if ( p e. Prime , ( log ` p ) , 0 ) e. CC /\ ( p e. CC /\ p =/= 0 ) ) -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) = ( ( (Λ ` p ) / p ) - ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) )
52	45, 49, 50, 51	syl3anc 1326	. . . . . . . . . 10 |- ( p e. NN -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) = ( ( (Λ ` p ) / p ) - ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) )
53	18, 52	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) = ( ( (Λ ` p ) / p ) - ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) )
54	53	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) = sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) / p ) - ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) )
55	21	recnd 10068	. . . . . . . . . 10 |- ( p e. NN -> ( (Λ ` p ) / p ) e. CC )
56	18, 55	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( (Λ ` p ) / p ) e. CC )
57	48, 36	rerpdivcld 11903	. . . . . . . . . . 11 |- ( p e. NN -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) e. RR )
58	57	recnd 10068	. . . . . . . . . 10 |- ( p e. NN -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) e. CC )
59	18, 58	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) e. CC )
60	14, 56, 59	fsumsub 14520	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) / p ) - ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) = ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) )
61		inss2 3834	. . . . . . . . . . . 12 |- ( Prime i^i T ) C_ T
62		sslin 3839	. . . . . . . . . . . 12 |- ( ( Prime i^i T ) C_ T -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( ( 1 ... ( |_ ` x ) ) i^i T ) )
63	61, 62	mp1i 13	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( ( 1 ... ( |_ ` x ) ) i^i T ) )
64	35, 58	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) e. CC )
65		eldif 3584	. . . . . . . . . . . . . . . 16 |- ( p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) <-> ( p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) /\ -. p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) )
66		incom 3805	. . . . . . . . . . . . . . . . . . . . 21 |- ( Prime i^i T ) = ( T i^i Prime )
67	66	ineq2i 3811	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) = ( ( 1 ... ( |_ ` x ) ) i^i ( T i^i Prime ) )
68		inass 3823	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1 ... ( |_ ` x ) ) i^i T ) i^i Prime ) = ( ( 1 ... ( |_ ` x ) ) i^i ( T i^i Prime ) )
69	67, 68	eqtr4i 2647	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) = ( ( ( 1 ... ( |_ ` x ) ) i^i T ) i^i Prime )
70	69	elin2 3801	. . . . . . . . . . . . . . . . . 18 |- ( p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) <-> ( p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) /\ p e. Prime ) )
71	70	simplbi2 655	. . . . . . . . . . . . . . . . 17 |- ( p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) -> ( p e. Prime -> p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) )
72	71	con3dimp 457	. . . . . . . . . . . . . . . 16 |- ( ( p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) /\ -. p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> -. p e. Prime )
73	65, 72	sylbi 207	. . . . . . . . . . . . . . 15 |- ( p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> -. p e. Prime )
74	73	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> -. p e. Prime )
75	74	iffalsed 4097	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> if ( p e. Prime , ( log ` p ) , 0 ) = 0 )
76	75	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = ( 0 / p ) )
77		eldifi 3732	. . . . . . . . . . . . . 14 |- ( p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) )
78	77, 18	sylan2 491	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> p e. NN )
79		div0 10715	. . . . . . . . . . . . . 14 |- ( ( p e. CC /\ p =/= 0 ) -> ( 0 / p ) = 0 )
80	50, 79	syl 17	. . . . . . . . . . . . 13 |- ( p e. NN -> ( 0 / p ) = 0 )
81	78, 80	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> ( 0 / p ) = 0 )
82	76, 81	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( ( 1 ... ( |_ ` x ) ) i^i T ) \ ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) ) -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = 0 )
83	63, 64, 82, 14	fsumss 14456	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) )
84		inss2 3834	. . . . . . . . . . . . . . 15 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( Prime i^i T )
85		inss1 3833	. . . . . . . . . . . . . . 15 |- ( Prime i^i T ) C_ Prime
86	84, 85	sstri 3612	. . . . . . . . . . . . . 14 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ Prime
87	86, 33	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> p e. Prime )
88	87	iftrued 4094	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> if ( p e. Prime , ( log ` p ) , 0 ) = ( log ` p ) )
89	88	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = ( ( log ` p ) / p ) )
90	89	sumeq2dv 14433	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) )
91	83, 90	eqtr3d 2658	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) = sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) )
92	91	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( if ( p e. Prime , ( log ` p ) , 0 ) / p ) ) = ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) )
93	54, 60, 92	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) = ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) )
94	93	oveq2d 6666	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) = ( ( phi ` N ) x. ( sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) - sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) ) )
95	25, 42, 28	nnncan2d 10427	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) - ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) = ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) ) )
96	44, 94, 95	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) = ( ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) - ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) )
97	96	mpteq2dva 4744	. . . 4 |- ( ph -> ( x e. RR+ |-> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) ) = ( x e. RR+ |-> ( ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) - ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) ) )
98	19, 48	resubcld 10458	. . . . . . . . 9 |- ( p e. NN -> ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) e. RR )
99	98, 36	rerpdivcld 11903	. . . . . . . 8 |- ( p e. NN -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. RR )
100	18, 99	syl 17	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. RR )
101	14, 100	fsumrecl 14465	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. RR )
102	101	recnd 10068	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. CC )
103		rpssre 11843	. . . . . 6 |- RR+ C_ RR
104	8	nncnd 11036	. . . . . 6 |- ( ph -> ( phi ` N ) e. CC )
105		o1const 14350	. . . . . 6 |- ( ( RR+ C_ RR /\ ( phi ` N ) e. CC ) -> ( x e. RR+ |-> ( phi ` N ) ) e. O(1) )
106	103, 104, 105	sylancr 695	. . . . 5 |- ( ph -> ( x e. RR+ |-> ( phi ` N ) ) e. O(1) )
107	103	a1i 11	. . . . . 6 |- ( ph -> RR+ C_ RR )
108		1red 10055	. . . . . 6 |- ( ph -> 1 e. RR )
109		2re 11090	. . . . . . 7 |- 2 e. RR
110	109	a1i 11	. . . . . 6 |- ( ph -> 2 e. RR )
111		breq1 4656	. . . . . . . . . . . . . 14 |- ( ( log ` p ) = if ( p e. Prime , ( log ` p ) , 0 ) -> ( ( log ` p ) <_ (Λ ` p ) <-> if ( p e. Prime , ( log ` p ) , 0 ) <_ (Λ ` p ) ) )
112		breq1 4656	. . . . . . . . . . . . . 14 |- ( 0 = if ( p e. Prime , ( log ` p ) , 0 ) -> ( 0 <_ (Λ ` p ) <-> if ( p e. Prime , ( log ` p ) , 0 ) <_ (Λ ` p ) ) )
113	37	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( p e. NN /\ p e. Prime ) -> ( log ` p ) e. RR )
114		vmaprm 24843	. . . . . . . . . . . . . . . . 17 |- ( p e. Prime -> (Λ ` p ) = ( log ` p ) )
115	114	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( p e. NN /\ p e. Prime ) -> (Λ ` p ) = ( log ` p ) )
116	115	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( p e. NN /\ p e. Prime ) -> ( log ` p ) = (Λ ` p ) )
117		eqle 10139	. . . . . . . . . . . . . . 15 |- ( ( ( log ` p ) e. RR /\ ( log ` p ) = (Λ ` p ) ) -> ( log ` p ) <_ (Λ ` p ) )
118	113, 116, 117	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( p e. NN /\ p e. Prime ) -> ( log ` p ) <_ (Λ ` p ) )
119		vmage0 24847	. . . . . . . . . . . . . . 15 |- ( p e. NN -> 0 <_ (Λ ` p ) )
120	119	adantr 481	. . . . . . . . . . . . . 14 |- ( ( p e. NN /\ -. p e. Prime ) -> 0 <_ (Λ ` p ) )
121	111, 112, 118, 120	ifbothda 4123	. . . . . . . . . . . . 13 |- ( p e. NN -> if ( p e. Prime , ( log ` p ) , 0 ) <_ (Λ ` p ) )
122	19, 48	subge0d 10617	. . . . . . . . . . . . 13 |- ( p e. NN -> ( 0 <_ ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) <-> if ( p e. Prime , ( log ` p ) , 0 ) <_ (Λ ` p ) ) )
123	121, 122	mpbird 247	. . . . . . . . . . . 12 |- ( p e. NN -> 0 <_ ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) )
124	98, 36, 123	divge0d 11912	. . . . . . . . . . 11 |- ( p e. NN -> 0 <_ ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
125	18, 124	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ) -> 0 <_ ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
126	14, 100, 125	fsumge0 14527	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> 0 <_ sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
127	101, 126	absidd 14161	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) = sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
128	17	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( 1 ... ( |_ ` x ) ) ) -> p e. NN )
129	128, 99	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( 1 ... ( |_ ` x ) ) ) -> ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. RR )
130	11, 129	fsumrecl 14465	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) e. RR )
131	109	a1i 11	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> 2 e. RR )
132	128, 124	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ p e. ( 1 ... ( |_ ` x ) ) ) -> 0 <_ ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
133	12	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> ( ( 1 ... ( |_ ` x ) ) i^i T ) C_ ( 1 ... ( |_ ` x ) ) )
134	11, 129, 132, 133	fsumless 14528	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) <_ sum_ p e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) )
135	107	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ x e. RR+ ) -> x e. RR )
136	135	flcld 12599	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> ( |_ ` x ) e. ZZ )
137		rplogsumlem2 25174	. . . . . . . . . 10 |- ( ( |_ ` x ) e. ZZ -> sum_ p e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) <_ 2 )
138	136, 137	syl 17	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( 1 ... ( |_ ` x ) ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) <_ 2 )
139	101, 130, 131, 134, 138	letrd 10194	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) <_ 2 )
140	127, 139	eqbrtrd 4675	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( abs ` sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) <_ 2 )
141	140	adantrr 753	. . . . . 6 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) <_ 2 )
142	107, 102, 108, 110, 141	elo1d 14267	. . . . 5 |- ( ph -> ( x e. RR+ |-> sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) e. O(1) )
143	10, 102, 106, 142	o1mul2 14355	. . . 4 |- ( ph -> ( x e. RR+ |-> ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( ( (Λ ` p ) - if ( p e. Prime , ( log ` p ) , 0 ) ) / p ) ) ) e. O(1) )
144	97, 143	eqeltrrd 2702	. . 3 |- ( ph -> ( x e. RR+ |-> ( ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) - ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) ) e. O(1) )
145	29, 43, 144	o1dif 14360	. 2 |- ( ph -> ( ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i T ) ( (Λ ` p ) / p ) ) - ( log ` x ) ) ) e. O(1) <-> ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) e. O(1) ) )
146	7, 145	mpbid 222	1 |- ( ph -> ( x e. RR+ |-> ( ( ( phi ` N ) x. sum_ p e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` p ) / p ) ) - ( log ` x ) ) ) e. O(1) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   ZZcz 11377   RR+crp 11832   ...cfz 12326   |_cfl 12591   abscabs 13974   O(1)co1 14217   sum_csu 14416   Primecprime 15385   phicphi 15469  Unitcui 18639   ZRHomczrh 19848  ℤ/nℤczn 19851   logclog 24301  Λcvma 24818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-o1 14221  df-lo1 14222  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-numer 15443  df-denom 15444  df-phi 15471  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-gim 17701  df-ga 17723  df-cntz 17750  df-oppg 17776  df-od 17948  df-gex 17949  df-pgp 17950  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-cyg 18280  df-dprd 18394  df-dpj 18395  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046  df-log 24303  df-cxp 24304  df-em 24719  df-cht 24823  df-vma 24824  df-chp 24825  df-ppi 24826  df-mu 24827  df-dchr 24958

This theorem is referenced by:  dirith2  25217