Theorem dchrvmasumiflem2 25191

Description: Lemma for dchrvmasum 25214. (Contributed by Mario Carneiro, 5-May-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	|- Z = (ℤ/nℤ ` N )
rpvmasum.l	|- L = ( ZRHom ` Z )
rpvmasum.a	|- ( ph -> N e. NN )
rpvmasum.g	|- G = (DChr ` N )
rpvmasum.d	|- D = ( Base ` G )
rpvmasum.1	|- .1. = ( 0g ` G )
dchrisum.b	|- ( ph -> X e. D )
dchrisum.n1	|- ( ph -> X =/= .1. )
dchrvmasumif.f	|- F = ( a e. NN |-> ( ( X ` ( L ` a ) ) / a ) )
dchrvmasumif.c	|- ( ph -> C e. ( 0 [,) +oo ) )
dchrvmasumif.s	|- ( ph -> seq 1 ( + , F ) ~~> S )
dchrvmasumif.1	|- ( ph -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) <_ ( C / y ) )
dchrvmasumif.g	|- K = ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) )
dchrvmasumif.e	|- ( ph -> E e. ( 0 [,) +oo ) )
dchrvmasumif.t	|- ( ph -> seq 1 ( + , K ) ~~> T )
dchrvmasumif.2	|- ( ph -> A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) <_ ( E x. ( ( log ` y ) / y ) ) )

Assertion

Ref	Expression
dchrvmasumiflem2	|- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) e. O(1) )

Distinct variable groups:   x, n, y, .1.   C, n, x, y   n, F, x, y   x, a, y   x, E, y   y, K   n, N, x, y   ph, n, x   T, n, x, y   S, n, x, y   n, Z, x, y   D, n, x, y   n, a, L, x, y   X, a, n, x, y

Allowed substitution hints:   ph( y, a)   C( a)   D( a)   S( a)   T( a)   .1. ( a)   E( n, a)   F( a)   G( x, y, n, a)   K( x, n, a)   N( a)   Z( a)

Proof of Theorem dchrvmasumiflem2

Dummy variables k d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 10055	. 2 |- ( ph -> 1 e. RR )
2		fzfid 12772	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> ( 1 ... ( |_ ` x ) ) e. Fin )
3		rpvmasum.g	. . . . . . . 8 |- G = (DChr ` N )
4		rpvmasum.z	. . . . . . . 8 |- Z = (ℤ/nℤ ` N )
5		rpvmasum.d	. . . . . . . 8 |- D = ( Base ` G )
6		rpvmasum.l	. . . . . . . 8 |- L = ( ZRHom ` Z )
7		dchrisum.b	. . . . . . . . 9 |- ( ph -> X e. D )
8	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> X e. D )
9		elfzelz 12342	. . . . . . . . 9 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. ZZ )
10	9	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. ZZ )
11	3, 4, 5, 6, 8, 10	dchrzrhcl 24970	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( X ` ( L ` d ) ) e. CC )
12		elfznn 12370	. . . . . . . . . . . 12 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. NN )
13	12	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> d e. NN )
14		mucl 24867	. . . . . . . . . . 11 |- ( d e. NN -> ( mmu ` d ) e. ZZ )
15	13, 14	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. ZZ )
16	15	zred 11482	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( mmu ` d ) e. RR )
17	16, 13	nndivred 11069	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. RR )
18	17	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( mmu ` d ) / d ) e. CC )
19	11, 18	mulcld 10060	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) e. CC )
20	2, 19	fsumcl 14464	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) e. CC )
21		dchrvmasumif.s	. . . . . . 7 |- ( ph -> seq 1 ( + , F ) ~~> S )
22		climcl 14230	. . . . . . 7 |- ( seq 1 ( + , F ) ~~> S -> S e. CC )
23	21, 22	syl 17	. . . . . 6 |- ( ph -> S e. CC )
24	23	adantr 481	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> S e. CC )
25	20, 24	mulcld 10060	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) e. CC )
26		0cnd 10033	. . . . . 6 |- ( ( ph /\ S = 0 ) -> 0 e. CC )
27		df-ne 2795	. . . . . . 7 |- ( S =/= 0 <-> -. S = 0 )
28		dchrvmasumif.t	. . . . . . . . . 10 |- ( ph -> seq 1 ( + , K ) ~~> T )
29		climcl 14230	. . . . . . . . . 10 |- ( seq 1 ( + , K ) ~~> T -> T e. CC )
30	28, 29	syl 17	. . . . . . . . 9 |- ( ph -> T e. CC )
31	30	adantr 481	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> T e. CC )
32	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> S e. CC )
33		simpr 477	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> S =/= 0 )
34	31, 32, 33	divcld 10801	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> ( T / S ) e. CC )
35	27, 34	sylan2br 493	. . . . . 6 |- ( ( ph /\ -. S = 0 ) -> ( T / S ) e. CC )
36	26, 35	ifclda 4120	. . . . 5 |- ( ph -> if ( S = 0 , 0 , ( T / S ) ) e. CC )
37	36	adantr 481	. . . 4 |- ( ( ph /\ x e. RR+ ) -> if ( S = 0 , 0 , ( T / S ) ) e. CC )
38		rpvmasum.a	. . . . 5 |- ( ph -> N e. NN )
39		rpvmasum.1	. . . . 5 |- .1. = ( 0g ` G )
40		dchrisum.n1	. . . . 5 |- ( ph -> X =/= .1. )
41		dchrvmasumif.f	. . . . 5 |- F = ( a e. NN |-> ( ( X ` ( L ` a ) ) / a ) )
42		dchrvmasumif.c	. . . . 5 |- ( ph -> C e. ( 0 [,) +oo ) )
43		dchrvmasumif.1	. . . . 5 |- ( ph -> A. y e. ( 1 [,) +oo ) ( abs ` ( ( seq 1 ( + , F ) ` ( |_ ` y ) ) - S ) ) <_ ( C / y ) )
44	4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43	dchrmusum2 25183	. . . 4 |- ( ph -> ( x e. RR+ |-> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) ) e. O(1) )
45		rpssre 11843	. . . . 5 |- RR+ C_ RR
46		o1const 14350	. . . . 5 |- ( ( RR+ C_ RR /\ if ( S = 0 , 0 , ( T / S ) ) e. CC ) -> ( x e. RR+ |-> if ( S = 0 , 0 , ( T / S ) ) ) e. O(1) )
47	45, 36, 46	sylancr 695	. . . 4 |- ( ph -> ( x e. RR+ |-> if ( S = 0 , 0 , ( T / S ) ) ) e. O(1) )
48	25, 37, 44, 47	o1mul2 14355	. . 3 |- ( ph -> ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) e. O(1) )
49		fzfid 12772	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( 1 ... ( |_ ` ( x / d ) ) ) e. Fin )
50	8	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> X e. D )
51		elfzelz 12342	. . . . . . . . . 10 |- ( k e. ( 1 ... ( |_ ` ( x / d ) ) ) -> k e. ZZ )
52	51	adantl 482	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> k e. ZZ )
53	3, 4, 5, 6, 50, 52	dchrzrhcl 24970	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( X ` ( L ` k ) ) e. CC )
54		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR+ ) -> x e. RR+ )
55	12	nnrpd 11870	. . . . . . . . . . . . 13 |- ( d e. ( 1 ... ( |_ ` x ) ) -> d e. RR+ )
56		rpdivcl 11856	. . . . . . . . . . . . 13 |- ( ( x e. RR+ /\ d e. RR+ ) -> ( x / d ) e. RR+ )
57	54, 55, 56	syl2an 494	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( x / d ) e. RR+ )
58		elfznn 12370	. . . . . . . . . . . . 13 |- ( k e. ( 1 ... ( |_ ` ( x / d ) ) ) -> k e. NN )
59	58	nnrpd 11870	. . . . . . . . . . . 12 |- ( k e. ( 1 ... ( |_ ` ( x / d ) ) ) -> k e. RR+ )
60		ifcl 4130	. . . . . . . . . . . 12 |- ( ( ( x / d ) e. RR+ /\ k e. RR+ ) -> if ( S = 0 , ( x / d ) , k ) e. RR+ )
61	57, 59, 60	syl2an 494	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> if ( S = 0 , ( x / d ) , k ) e. RR+ )
62	61	relogcld 24369	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( log ` if ( S = 0 , ( x / d ) , k ) ) e. RR )
63	58	adantl 482	. . . . . . . . . 10 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> k e. NN )
64	62, 63	nndivred 11069	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) e. RR )
65	64	recnd 10068	. . . . . . . 8 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) e. CC )
66	53, 65	mulcld 10060	. . . . . . 7 |- ( ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) /\ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ) -> ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) e. CC )
67	49, 66	fsumcl 14464	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) e. CC )
68	19, 67	mulcld 10060	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) e. CC )
69	2, 68	fsumcl 14464	. . . 4 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) e. CC )
70	25, 37	mulcld 10060	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) e. CC )
71		0cn 10032	. . . . . . . . . 10 |- 0 e. CC
72	30	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> T e. CC )
73		ifcl 4130	. . . . . . . . . 10 |- ( ( 0 e. CC /\ T e. CC ) -> if ( S = 0 , 0 , T ) e. CC )
74	71, 72, 73	sylancr 695	. . . . . . . . 9 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> if ( S = 0 , 0 , T ) e. CC )
75	19, 67, 74	subdid 10486	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = ( ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) ) )
76	75	sumeq2dv 14433	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) ) )
77	19, 74	mulcld 10060	. . . . . . . 8 |- ( ( ( ph /\ x e. RR+ ) /\ d e. ( 1 ... ( |_ ` x ) ) ) -> ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) e. CC )
78	2, 68, 77	fsumsub 14520	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) ) )
79	20, 24, 37	mulassd 10063	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( S x. if ( S = 0 , 0 , ( T / S ) ) ) ) )
80		ovif2 6738	. . . . . . . . . . . 12 |- ( S x. if ( S = 0 , 0 , ( T / S ) ) ) = if ( S = 0 , ( S x. 0 ) , ( S x. ( T / S ) ) )
81	23	mul01d 10235	. . . . . . . . . . . . . 14 |- ( ph -> ( S x. 0 ) = 0 )
82	81	ifeq1d 4104	. . . . . . . . . . . . 13 |- ( ph -> if ( S = 0 , ( S x. 0 ) , ( S x. ( T / S ) ) ) = if ( S = 0 , 0 , ( S x. ( T / S ) ) ) )
83	31, 32, 33	divcan2d 10803	. . . . . . . . . . . . . . 15 |- ( ( ph /\ S =/= 0 ) -> ( S x. ( T / S ) ) = T )
84	27, 83	sylan2br 493	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. S = 0 ) -> ( S x. ( T / S ) ) = T )
85	84	ifeq2da 4117	. . . . . . . . . . . . 13 |- ( ph -> if ( S = 0 , 0 , ( S x. ( T / S ) ) ) = if ( S = 0 , 0 , T ) )
86	82, 85	eqtrd 2656	. . . . . . . . . . . 12 |- ( ph -> if ( S = 0 , ( S x. 0 ) , ( S x. ( T / S ) ) ) = if ( S = 0 , 0 , T ) )
87	80, 86	syl5eq 2668	. . . . . . . . . . 11 |- ( ph -> ( S x. if ( S = 0 , 0 , ( T / S ) ) ) = if ( S = 0 , 0 , T ) )
88	87	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> ( S x. if ( S = 0 , 0 , ( T / S ) ) ) = if ( S = 0 , 0 , T ) )
89	88	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( S x. if ( S = 0 , 0 , ( T / S ) ) ) ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) )
90	71, 30, 73	sylancr 695	. . . . . . . . . . 11 |- ( ph -> if ( S = 0 , 0 , T ) e. CC )
91	90	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> if ( S = 0 , 0 , T ) e. CC )
92	2, 91, 19	fsummulc1 14517	. . . . . . . . 9 |- ( ( ph /\ x e. RR+ ) -> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) )
93	79, 89, 92	3eqtrrd 2661	. . . . . . . 8 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) = ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) )
94	93	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ x e. RR+ ) -> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. if ( S = 0 , 0 , T ) ) ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) )
95	76, 78, 94	3eqtrd 2660	. . . . . 6 |- ( ( ph /\ x e. RR+ ) -> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) = ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) )
96	95	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) ) = ( x e. RR+ |-> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) ) )
97		dchrvmasumif.g	. . . . . 6 |- K = ( a e. NN |-> ( ( X ` ( L ` a ) ) x. ( ( log ` a ) / a ) ) )
98		dchrvmasumif.e	. . . . . 6 |- ( ph -> E e. ( 0 [,) +oo ) )
99		dchrvmasumif.2	. . . . . 6 |- ( ph -> A. y e. ( 3 [,) +oo ) ( abs ` ( ( seq 1 ( + , K ) ` ( |_ ` y ) ) - T ) ) <_ ( E x. ( ( log ` y ) / y ) ) )
100	4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43, 97, 98, 28, 99	dchrvmasumiflem1 25190	. . . . 5 |- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. ( sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) - if ( S = 0 , 0 , T ) ) ) ) e. O(1) )
101	96, 100	eqeltrrd 2702	. . . 4 |- ( ph -> ( x e. RR+ |-> ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) - ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) ) e. O(1) )
102	69, 70, 101	o1dif 14360	. . 3 |- ( ph -> ( ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) e. O(1) <-> ( x e. RR+ |-> ( ( sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. S ) x. if ( S = 0 , 0 , ( T / S ) ) ) ) e. O(1) ) )
103	48, 102	mpbird 247	. 2 |- ( ph -> ( x e. RR+ |-> sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) e. O(1) )
104	7	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> X e. D )
105		elfzelz 12342	. . . . . . 7 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. ZZ )
106	105	adantl 482	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. ZZ )
107	3, 4, 5, 6, 104, 106	dchrzrhcl 24970	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( X ` ( L ` n ) ) e. CC )
108		elfznn 12370	. . . . . . . 8 |- ( n e. ( 1 ... ( |_ ` x ) ) -> n e. NN )
109	108	adantl 482	. . . . . . 7 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> n e. NN )
110		vmacl 24844	. . . . . . . 8 |- ( n e. NN -> (Λ ` n ) e. RR )
111		nndivre 11056	. . . . . . . 8 |- ( ( (Λ ` n ) e. RR /\ n e. NN ) -> ( (Λ ` n ) / n ) e. RR )
112	110, 111	mpancom 703	. . . . . . 7 |- ( n e. NN -> ( (Λ ` n ) / n ) e. RR )
113	109, 112	syl 17	. . . . . 6 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) / n ) e. RR )
114	113	recnd 10068	. . . . 5 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( (Λ ` n ) / n ) e. CC )
115	107, 114	mulcld 10060	. . . 4 |- ( ( ( ph /\ x e. RR+ ) /\ n e. ( 1 ... ( |_ ` x ) ) ) -> ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) e. CC )
116	2, 115	fsumcl 14464	. . 3 |- ( ( ph /\ x e. RR+ ) -> sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) e. CC )
117		relogcl 24322	. . . . . 6 |- ( x e. RR+ -> ( log ` x ) e. RR )
118	117	adantl 482	. . . . 5 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. RR )
119	118	recnd 10068	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( log ` x ) e. CC )
120		ifcl 4130	. . . 4 |- ( ( ( log ` x ) e. CC /\ 0 e. CC ) -> if ( S = 0 , ( log ` x ) , 0 ) e. CC )
121	119, 71, 120	sylancl 694	. . 3 |- ( ( ph /\ x e. RR+ ) -> if ( S = 0 , ( log ` x ) , 0 ) e. CC )
122	116, 121	addcld 10059	. 2 |- ( ( ph /\ x e. RR+ ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) e. CC )
123	122	abscld 14175	. . . 4 |- ( ( ph /\ x e. RR+ ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) e. RR )
124	123	adantrr 753	. . 3 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) e. RR )
125	38	adantr 481	. . . . 5 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> N e. NN )
126	7	adantr 481	. . . . 5 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> X e. D )
127	40	adantr 481	. . . . 5 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> X =/= .1. )
128		simprl 794	. . . . 5 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> x e. RR+ )
129		simprr 796	. . . . 5 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> 1 <_ x )
130	4, 6, 125, 3, 5, 39, 126, 127, 128, 129	dchrvmasum2if 25186	. . . 4 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) = sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) )
131	130	fveq2d 6195	. . 3 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) = ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) )
132		eqle 10139	. . 3 |- ( ( ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) e. RR /\ ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) = ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) )
133	124, 131, 132	syl2anc 693	. 2 |- ( ( ph /\ ( x e. RR+ /\ 1 <_ x ) ) -> ( abs ` ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) <_ ( abs ` sum_ d e. ( 1 ... ( |_ ` x ) ) ( ( ( X ` ( L ` d ) ) x. ( ( mmu ` d ) / d ) ) x. sum_ k e. ( 1 ... ( |_ ` ( x / d ) ) ) ( ( X ` ( L ` k ) ) x. ( ( log ` if ( S = 0 , ( x / d ) , k ) ) / k ) ) ) ) )
134	1, 103, 69, 122, 133	o1le 14383	1 |- ( ph -> ( x e. RR+ |-> ( sum_ n e. ( 1 ... ( |_ ` x ) ) ( ( X ` ( L ` n ) ) x. ( (Λ ` n ) / n ) ) + if ( S = 0 , ( log ` x ) , 0 ) ) ) e. O(1) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   3c3 11071   ZZcz 11377   RR+crp 11832   [,)cico 12177   ...cfz 12326   |_cfl 12591   seqcseq 12801   abscabs 13974   ~~> cli 14215   O(1)co1 14217   sum_csu 14416   Basecbs 15857   0gc0g 16100   ZRHomczrh 19848  ℤ/nℤczn 19851   logclog 24301  Λcvma 24818   mmucmu 24821  DChrcdchr 24957

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-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-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-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  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-cntz 17750  df-od 17948  df-cmn 18195  df-abl 18196  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-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-em 24719  df-vma 24824  df-mu 24827  df-dchr 24958

This theorem is referenced by:  dchrvmasumif  25192