Step |
Hyp |
Ref |
Expression |
1 |
|
fsumvma2.1 |
|- ( x = ( p ^ k ) -> B = C ) |
2 |
|
fsumvma2.2 |
|- ( ph -> A e. RR ) |
3 |
|
fsumvma2.3 |
|- ( ( ph /\ x e. ( 1 ... ( |_ ` A ) ) ) -> B e. CC ) |
4 |
|
fsumvma2.4 |
|- ( ( ph /\ ( x e. ( 1 ... ( |_ ` A ) ) /\ ( Lam ` x ) = 0 ) ) -> B = 0 ) |
5 |
|
fzfid |
|- ( ph -> ( 1 ... ( |_ ` A ) ) e. Fin ) |
6 |
|
fz1ssnn |
|- ( 1 ... ( |_ ` A ) ) C_ NN |
7 |
6
|
a1i |
|- ( ph -> ( 1 ... ( |_ ` A ) ) C_ NN ) |
8 |
|
ppifi |
|- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) e. Fin ) |
9 |
2 8
|
syl |
|- ( ph -> ( ( 0 [,] A ) i^i Prime ) e. Fin ) |
10 |
|
elinel2 |
|- ( p e. ( ( 0 [,] A ) i^i Prime ) -> p e. Prime ) |
11 |
|
elfznn |
|- ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) -> k e. NN ) |
12 |
10 11
|
anim12i |
|- ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) -> ( p e. Prime /\ k e. NN ) ) |
13 |
12
|
pm4.71ri |
|- ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) ) |
14 |
2
|
adantr |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> A e. RR ) |
15 |
|
prmnn |
|- ( p e. Prime -> p e. NN ) |
16 |
15
|
ad2antrl |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. NN ) |
17 |
|
nnnn0 |
|- ( k e. NN -> k e. NN0 ) |
18 |
17
|
ad2antll |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. NN0 ) |
19 |
16 18
|
nnexpcld |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. NN ) |
20 |
19
|
nnzd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. ZZ ) |
21 |
|
flge |
|- ( ( A e. RR /\ ( p ^ k ) e. ZZ ) -> ( ( p ^ k ) <_ A <-> ( p ^ k ) <_ ( |_ ` A ) ) ) |
22 |
14 20 21
|
syl2anc |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A <-> ( p ^ k ) <_ ( |_ ` A ) ) ) |
23 |
|
simplrl |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. Prime ) |
24 |
23 15
|
syl |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. NN ) |
25 |
24
|
nnrpd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. RR+ ) |
26 |
|
simplrr |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. NN ) |
27 |
26
|
nnzd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. ZZ ) |
28 |
|
relogexp |
|- ( ( p e. RR+ /\ k e. ZZ ) -> ( log ` ( p ^ k ) ) = ( k x. ( log ` p ) ) ) |
29 |
25 27 28
|
syl2anc |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` ( p ^ k ) ) = ( k x. ( log ` p ) ) ) |
30 |
29
|
breq1d |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` ( p ^ k ) ) <_ ( log ` A ) <-> ( k x. ( log ` p ) ) <_ ( log ` A ) ) ) |
31 |
26
|
nnred |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. RR ) |
32 |
14
|
adantr |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> A e. RR ) |
33 |
|
0red |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 e. RR ) |
34 |
16
|
nnred |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. RR ) |
35 |
34
|
adantr |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p e. RR ) |
36 |
24
|
nngt0d |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 < p ) |
37 |
|
0red |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 0 e. RR ) |
38 |
16
|
nnnn0d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. NN0 ) |
39 |
38
|
nn0ge0d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 0 <_ p ) |
40 |
|
elicc2 |
|- ( ( 0 e. RR /\ A e. RR ) -> ( p e. ( 0 [,] A ) <-> ( p e. RR /\ 0 <_ p /\ p <_ A ) ) ) |
41 |
|
df-3an |
|- ( ( p e. RR /\ 0 <_ p /\ p <_ A ) <-> ( ( p e. RR /\ 0 <_ p ) /\ p <_ A ) ) |
42 |
40 41
|
bitrdi |
|- ( ( 0 e. RR /\ A e. RR ) -> ( p e. ( 0 [,] A ) <-> ( ( p e. RR /\ 0 <_ p ) /\ p <_ A ) ) ) |
43 |
42
|
baibd |
|- ( ( ( 0 e. RR /\ A e. RR ) /\ ( p e. RR /\ 0 <_ p ) ) -> ( p e. ( 0 [,] A ) <-> p <_ A ) ) |
44 |
37 14 34 39 43
|
syl22anc |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p e. ( 0 [,] A ) <-> p <_ A ) ) |
45 |
44
|
biimpa |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> p <_ A ) |
46 |
33 35 32 36 45
|
ltletrd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> 0 < A ) |
47 |
32 46
|
elrpd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> A e. RR+ ) |
48 |
47
|
relogcld |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` A ) e. RR ) |
49 |
|
prmuz2 |
|- ( p e. Prime -> p e. ( ZZ>= ` 2 ) ) |
50 |
|
eluzelre |
|- ( p e. ( ZZ>= ` 2 ) -> p e. RR ) |
51 |
|
eluz2gt1 |
|- ( p e. ( ZZ>= ` 2 ) -> 1 < p ) |
52 |
50 51
|
rplogcld |
|- ( p e. ( ZZ>= ` 2 ) -> ( log ` p ) e. RR+ ) |
53 |
23 49 52
|
3syl |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( log ` p ) e. RR+ ) |
54 |
31 48 53
|
lemuldivd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( k x. ( log ` p ) ) <_ ( log ` A ) <-> k <_ ( ( log ` A ) / ( log ` p ) ) ) ) |
55 |
48 53
|
rerpdivcld |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` A ) / ( log ` p ) ) e. RR ) |
56 |
|
flge |
|- ( ( ( ( log ` A ) / ( log ` p ) ) e. RR /\ k e. ZZ ) -> ( k <_ ( ( log ` A ) / ( log ` p ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) |
57 |
55 27 56
|
syl2anc |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( k <_ ( ( log ` A ) / ( log ` p ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) |
58 |
30 54 57
|
3bitrd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( log ` ( p ^ k ) ) <_ ( log ` A ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) |
59 |
19
|
adantr |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( p ^ k ) e. NN ) |
60 |
59
|
nnrpd |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( p ^ k ) e. RR+ ) |
61 |
60 47
|
logled |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( p ^ k ) <_ A <-> ( log ` ( p ^ k ) ) <_ ( log ` A ) ) ) |
62 |
|
simprr |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. NN ) |
63 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
64 |
62 63
|
eleqtrdi |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> k e. ( ZZ>= ` 1 ) ) |
65 |
64
|
adantr |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> k e. ( ZZ>= ` 1 ) ) |
66 |
55
|
flcld |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) e. ZZ ) |
67 |
|
elfz5 |
|- ( ( k e. ( ZZ>= ` 1 ) /\ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) e. ZZ ) -> ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) |
68 |
65 66 67
|
syl2anc |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) <-> k <_ ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) |
69 |
58 61 68
|
3bitr4d |
|- ( ( ( ph /\ ( p e. Prime /\ k e. NN ) ) /\ p e. ( 0 [,] A ) ) -> ( ( p ^ k ) <_ A <-> k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) |
70 |
69
|
pm5.32da |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( 0 [,] A ) /\ ( p ^ k ) <_ A ) <-> ( p e. ( 0 [,] A ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) ) |
71 |
16
|
nncnd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p e. CC ) |
72 |
71
|
exp1d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ 1 ) = p ) |
73 |
16
|
nnge1d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> 1 <_ p ) |
74 |
34 73 64
|
leexp2ad |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ 1 ) <_ ( p ^ k ) ) |
75 |
72 74
|
eqbrtrrd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> p <_ ( p ^ k ) ) |
76 |
19
|
nnred |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. RR ) |
77 |
|
letr |
|- ( ( p e. RR /\ ( p ^ k ) e. RR /\ A e. RR ) -> ( ( p <_ ( p ^ k ) /\ ( p ^ k ) <_ A ) -> p <_ A ) ) |
78 |
34 76 14 77
|
syl3anc |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p <_ ( p ^ k ) /\ ( p ^ k ) <_ A ) -> p <_ A ) ) |
79 |
75 78
|
mpand |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A -> p <_ A ) ) |
80 |
79 44
|
sylibrd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A -> p e. ( 0 [,] A ) ) ) |
81 |
80
|
pm4.71rd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) <_ A <-> ( p e. ( 0 [,] A ) /\ ( p ^ k ) <_ A ) ) ) |
82 |
|
elin |
|- ( p e. ( ( 0 [,] A ) i^i Prime ) <-> ( p e. ( 0 [,] A ) /\ p e. Prime ) ) |
83 |
82
|
rbaib |
|- ( p e. Prime -> ( p e. ( ( 0 [,] A ) i^i Prime ) <-> p e. ( 0 [,] A ) ) ) |
84 |
83
|
ad2antrl |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p e. ( ( 0 [,] A ) i^i Prime ) <-> p e. ( 0 [,] A ) ) ) |
85 |
84
|
anbi1d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p e. ( 0 [,] A ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) ) |
86 |
70 81 85
|
3bitr4rd |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p ^ k ) <_ A ) ) |
87 |
19 63
|
eleqtrdi |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( p ^ k ) e. ( ZZ>= ` 1 ) ) |
88 |
14
|
flcld |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( |_ ` A ) e. ZZ ) |
89 |
|
elfz5 |
|- ( ( ( p ^ k ) e. ( ZZ>= ` 1 ) /\ ( |_ ` A ) e. ZZ ) -> ( ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) <-> ( p ^ k ) <_ ( |_ ` A ) ) ) |
90 |
87 88 89
|
syl2anc |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) <-> ( p ^ k ) <_ ( |_ ` A ) ) ) |
91 |
22 86 90
|
3bitr4d |
|- ( ( ph /\ ( p e. Prime /\ k e. NN ) ) -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) ) |
92 |
91
|
pm5.32da |
|- ( ph -> ( ( ( p e. Prime /\ k e. NN ) /\ ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) ) ) |
93 |
13 92
|
syl5bb |
|- ( ph -> ( ( p e. ( ( 0 [,] A ) i^i Prime ) /\ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) ) <-> ( ( p e. Prime /\ k e. NN ) /\ ( p ^ k ) e. ( 1 ... ( |_ ` A ) ) ) ) ) |
94 |
1 5 7 9 93 3 4
|
fsumvma |
|- ( ph -> sum_ x e. ( 1 ... ( |_ ` A ) ) B = sum_ p e. ( ( 0 [,] A ) i^i Prime ) sum_ k e. ( 1 ... ( |_ ` ( ( log ` A ) / ( log ` p ) ) ) ) C ) |