Step |
Hyp |
Ref |
Expression |
1 |
|
pntsval.1 |
|- S = ( a e. RR |-> sum_ i e. ( 1 ... ( |_ ` a ) ) ( ( Lam ` i ) x. ( ( log ` i ) + ( psi ` ( a / i ) ) ) ) ) |
2 |
1
|
pntsval |
|- ( A e. RR -> ( S ` A ) = sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( ( log ` n ) + ( psi ` ( A / n ) ) ) ) ) |
3 |
|
elfznn |
|- ( n e. ( 1 ... ( |_ ` A ) ) -> n e. NN ) |
4 |
3
|
adantl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> n e. NN ) |
5 |
|
vmacl |
|- ( n e. NN -> ( Lam ` n ) e. RR ) |
6 |
4 5
|
syl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( Lam ` n ) e. RR ) |
7 |
6
|
recnd |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( Lam ` n ) e. CC ) |
8 |
4
|
nnrpd |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> n e. RR+ ) |
9 |
8
|
relogcld |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` n ) e. RR ) |
10 |
9
|
recnd |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( log ` n ) e. CC ) |
11 |
|
simpl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> A e. RR ) |
12 |
11 4
|
nndivred |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( A / n ) e. RR ) |
13 |
|
chpcl |
|- ( ( A / n ) e. RR -> ( psi ` ( A / n ) ) e. RR ) |
14 |
12 13
|
syl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( psi ` ( A / n ) ) e. RR ) |
15 |
14
|
recnd |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( psi ` ( A / n ) ) e. CC ) |
16 |
7 10 15
|
adddid |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` n ) x. ( ( log ` n ) + ( psi ` ( A / n ) ) ) ) = ( ( ( Lam ` n ) x. ( log ` n ) ) + ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) ) |
17 |
16
|
sumeq2dv |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( ( log ` n ) + ( psi ` ( A / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) ) |
18 |
|
fveq2 |
|- ( n = m -> ( Lam ` n ) = ( Lam ` m ) ) |
19 |
|
oveq2 |
|- ( n = m -> ( A / n ) = ( A / m ) ) |
20 |
19
|
fveq2d |
|- ( n = m -> ( psi ` ( A / n ) ) = ( psi ` ( A / m ) ) ) |
21 |
18 20
|
oveq12d |
|- ( n = m -> ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) = ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) ) |
22 |
21
|
cbvsumv |
|- sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) = sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) |
23 |
|
fzfid |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 ... ( |_ ` ( A / m ) ) ) e. Fin ) |
24 |
|
elfznn |
|- ( m e. ( 1 ... ( |_ ` A ) ) -> m e. NN ) |
25 |
24
|
adantl |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> m e. NN ) |
26 |
|
vmacl |
|- ( m e. NN -> ( Lam ` m ) e. RR ) |
27 |
25 26
|
syl |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( Lam ` m ) e. RR ) |
28 |
27
|
recnd |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( Lam ` m ) e. CC ) |
29 |
|
elfznn |
|- ( k e. ( 1 ... ( |_ ` ( A / m ) ) ) -> k e. NN ) |
30 |
29
|
adantl |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> k e. NN ) |
31 |
|
vmacl |
|- ( k e. NN -> ( Lam ` k ) e. RR ) |
32 |
30 31
|
syl |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( Lam ` k ) e. RR ) |
33 |
32
|
recnd |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( Lam ` k ) e. CC ) |
34 |
23 28 33
|
fsummulc2 |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` m ) x. sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( Lam ` k ) ) = sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` k ) ) ) |
35 |
|
simpl |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> A e. RR ) |
36 |
35 25
|
nndivred |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( A / m ) e. RR ) |
37 |
|
chpval |
|- ( ( A / m ) e. RR -> ( psi ` ( A / m ) ) = sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( Lam ` k ) ) |
38 |
36 37
|
syl |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( psi ` ( A / m ) ) = sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( Lam ` k ) ) |
39 |
38
|
oveq2d |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) = ( ( Lam ` m ) x. sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( Lam ` k ) ) ) |
40 |
30
|
nncnd |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> k e. CC ) |
41 |
24
|
ad2antlr |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> m e. NN ) |
42 |
41
|
nncnd |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> m e. CC ) |
43 |
41
|
nnne0d |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> m =/= 0 ) |
44 |
40 42 43
|
divcan3d |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( ( m x. k ) / m ) = k ) |
45 |
44
|
fveq2d |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( Lam ` ( ( m x. k ) / m ) ) = ( Lam ` k ) ) |
46 |
45
|
oveq2d |
|- ( ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) /\ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ) -> ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) = ( ( Lam ` m ) x. ( Lam ` k ) ) ) |
47 |
46
|
sumeq2dv |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) = sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` k ) ) ) |
48 |
34 39 47
|
3eqtr4d |
|- ( ( A e. RR /\ m e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) = sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) ) |
49 |
48
|
sumeq2dv |
|- ( A e. RR -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) = sum_ m e. ( 1 ... ( |_ ` A ) ) sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) ) |
50 |
|
fvoveq1 |
|- ( n = ( m x. k ) -> ( Lam ` ( n / m ) ) = ( Lam ` ( ( m x. k ) / m ) ) ) |
51 |
50
|
oveq2d |
|- ( n = ( m x. k ) -> ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) = ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) ) |
52 |
|
id |
|- ( A e. RR -> A e. RR ) |
53 |
|
ssrab2 |
|- { y e. NN | y || n } C_ NN |
54 |
|
simpr |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> m e. { y e. NN | y || n } ) |
55 |
53 54
|
sseldi |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> m e. NN ) |
56 |
55 26
|
syl |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( Lam ` m ) e. RR ) |
57 |
|
dvdsdivcl |
|- ( ( n e. NN /\ m e. { y e. NN | y || n } ) -> ( n / m ) e. { y e. NN | y || n } ) |
58 |
4 57
|
sylan |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( n / m ) e. { y e. NN | y || n } ) |
59 |
53 58
|
sseldi |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( n / m ) e. NN ) |
60 |
|
vmacl |
|- ( ( n / m ) e. NN -> ( Lam ` ( n / m ) ) e. RR ) |
61 |
59 60
|
syl |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( Lam ` ( n / m ) ) e. RR ) |
62 |
56 61
|
remulcld |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) e. RR ) |
63 |
62
|
recnd |
|- ( ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) /\ m e. { y e. NN | y || n } ) -> ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) e. CC ) |
64 |
63
|
anasss |
|- ( ( A e. RR /\ ( n e. ( 1 ... ( |_ ` A ) ) /\ m e. { y e. NN | y || n } ) ) -> ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) e. CC ) |
65 |
51 52 64
|
dvdsflsumcom |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) = sum_ m e. ( 1 ... ( |_ ` A ) ) sum_ k e. ( 1 ... ( |_ ` ( A / m ) ) ) ( ( Lam ` m ) x. ( Lam ` ( ( m x. k ) / m ) ) ) ) |
66 |
49 65
|
eqtr4d |
|- ( A e. RR -> sum_ m e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` m ) x. ( psi ` ( A / m ) ) ) = sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) |
67 |
22 66
|
syl5eq |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) = sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) |
68 |
67
|
oveq2d |
|- ( A e. RR -> ( sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) ) |
69 |
|
fzfid |
|- ( A e. RR -> ( 1 ... ( |_ ` A ) ) e. Fin ) |
70 |
7 10
|
mulcld |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` n ) x. ( log ` n ) ) e. CC ) |
71 |
7 15
|
mulcld |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) e. CC ) |
72 |
69 70 71
|
fsumadd |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) ) |
73 |
|
fzfid |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> ( 1 ... n ) e. Fin ) |
74 |
|
dvdsssfz1 |
|- ( n e. NN -> { y e. NN | y || n } C_ ( 1 ... n ) ) |
75 |
4 74
|
syl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> { y e. NN | y || n } C_ ( 1 ... n ) ) |
76 |
73 75
|
ssfid |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> { y e. NN | y || n } e. Fin ) |
77 |
76 62
|
fsumrecl |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) e. RR ) |
78 |
77
|
recnd |
|- ( ( A e. RR /\ n e. ( 1 ... ( |_ ` A ) ) ) -> sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) e. CC ) |
79 |
69 70 78
|
fsumadd |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) = ( sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( Lam ` n ) x. ( log ` n ) ) + sum_ n e. ( 1 ... ( |_ ` A ) ) sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) ) |
80 |
68 72 79
|
3eqtr4d |
|- ( A e. RR -> sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + ( ( Lam ` n ) x. ( psi ` ( A / n ) ) ) ) = sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) ) |
81 |
2 17 80
|
3eqtrd |
|- ( A e. RR -> ( S ` A ) = sum_ n e. ( 1 ... ( |_ ` A ) ) ( ( ( Lam ` n ) x. ( log ` n ) ) + sum_ m e. { y e. NN | y || n } ( ( Lam ` m ) x. ( Lam ` ( n / m ) ) ) ) ) |