Step |
Hyp |
Ref |
Expression |
1 |
|
2lgslem2.n |
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) |
2 |
|
oveq1 |
|- ( P = ( ( 8 x. K ) + 7 ) -> ( P - 1 ) = ( ( ( 8 x. K ) + 7 ) - 1 ) ) |
3 |
2
|
oveq1d |
|- ( P = ( ( 8 x. K ) + 7 ) -> ( ( P - 1 ) / 2 ) = ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) ) |
4 |
|
fvoveq1 |
|- ( P = ( ( 8 x. K ) + 7 ) -> ( |_ ` ( P / 4 ) ) = ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) |
5 |
3 4
|
oveq12d |
|- ( P = ( ( 8 x. K ) + 7 ) -> ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) ) |
6 |
1 5
|
syl5eq |
|- ( P = ( ( 8 x. K ) + 7 ) -> N = ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) ) |
7 |
|
8nn0 |
|- 8 e. NN0 |
8 |
7
|
a1i |
|- ( K e. NN0 -> 8 e. NN0 ) |
9 |
|
id |
|- ( K e. NN0 -> K e. NN0 ) |
10 |
8 9
|
nn0mulcld |
|- ( K e. NN0 -> ( 8 x. K ) e. NN0 ) |
11 |
10
|
nn0cnd |
|- ( K e. NN0 -> ( 8 x. K ) e. CC ) |
12 |
|
7cn |
|- 7 e. CC |
13 |
12
|
a1i |
|- ( K e. NN0 -> 7 e. CC ) |
14 |
|
1cnd |
|- ( K e. NN0 -> 1 e. CC ) |
15 |
11 13 14
|
addsubassd |
|- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( 8 x. K ) + ( 7 - 1 ) ) ) |
16 |
|
4t2e8 |
|- ( 4 x. 2 ) = 8 |
17 |
16
|
eqcomi |
|- 8 = ( 4 x. 2 ) |
18 |
17
|
a1i |
|- ( K e. NN0 -> 8 = ( 4 x. 2 ) ) |
19 |
18
|
oveq1d |
|- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. 2 ) x. K ) ) |
20 |
|
4cn |
|- 4 e. CC |
21 |
20
|
a1i |
|- ( K e. NN0 -> 4 e. CC ) |
22 |
|
2cn |
|- 2 e. CC |
23 |
22
|
a1i |
|- ( K e. NN0 -> 2 e. CC ) |
24 |
|
nn0cn |
|- ( K e. NN0 -> K e. CC ) |
25 |
21 23 24
|
mul32d |
|- ( K e. NN0 -> ( ( 4 x. 2 ) x. K ) = ( ( 4 x. K ) x. 2 ) ) |
26 |
19 25
|
eqtrd |
|- ( K e. NN0 -> ( 8 x. K ) = ( ( 4 x. K ) x. 2 ) ) |
27 |
|
7m1e6 |
|- ( 7 - 1 ) = 6 |
28 |
27
|
a1i |
|- ( K e. NN0 -> ( 7 - 1 ) = 6 ) |
29 |
26 28
|
oveq12d |
|- ( K e. NN0 -> ( ( 8 x. K ) + ( 7 - 1 ) ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) ) |
30 |
15 29
|
eqtrd |
|- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) - 1 ) = ( ( ( 4 x. K ) x. 2 ) + 6 ) ) |
31 |
30
|
oveq1d |
|- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) ) |
32 |
|
4nn0 |
|- 4 e. NN0 |
33 |
32
|
a1i |
|- ( K e. NN0 -> 4 e. NN0 ) |
34 |
33 9
|
nn0mulcld |
|- ( K e. NN0 -> ( 4 x. K ) e. NN0 ) |
35 |
34
|
nn0cnd |
|- ( K e. NN0 -> ( 4 x. K ) e. CC ) |
36 |
35 23
|
mulcld |
|- ( K e. NN0 -> ( ( 4 x. K ) x. 2 ) e. CC ) |
37 |
|
6cn |
|- 6 e. CC |
38 |
37
|
a1i |
|- ( K e. NN0 -> 6 e. CC ) |
39 |
|
2rp |
|- 2 e. RR+ |
40 |
39
|
a1i |
|- ( K e. NN0 -> 2 e. RR+ ) |
41 |
40
|
rpcnne0d |
|- ( K e. NN0 -> ( 2 e. CC /\ 2 =/= 0 ) ) |
42 |
|
divdir |
|- ( ( ( ( 4 x. K ) x. 2 ) e. CC /\ 6 e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) ) |
43 |
36 38 41 42
|
syl3anc |
|- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) + 6 ) / 2 ) = ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) ) |
44 |
|
2ne0 |
|- 2 =/= 0 |
45 |
44
|
a1i |
|- ( K e. NN0 -> 2 =/= 0 ) |
46 |
35 23 45
|
divcan4d |
|- ( K e. NN0 -> ( ( ( 4 x. K ) x. 2 ) / 2 ) = ( 4 x. K ) ) |
47 |
|
3t2e6 |
|- ( 3 x. 2 ) = 6 |
48 |
47
|
eqcomi |
|- 6 = ( 3 x. 2 ) |
49 |
48
|
oveq1i |
|- ( 6 / 2 ) = ( ( 3 x. 2 ) / 2 ) |
50 |
|
3cn |
|- 3 e. CC |
51 |
50 22 44
|
divcan4i |
|- ( ( 3 x. 2 ) / 2 ) = 3 |
52 |
49 51
|
eqtri |
|- ( 6 / 2 ) = 3 |
53 |
52
|
a1i |
|- ( K e. NN0 -> ( 6 / 2 ) = 3 ) |
54 |
46 53
|
oveq12d |
|- ( K e. NN0 -> ( ( ( ( 4 x. K ) x. 2 ) / 2 ) + ( 6 / 2 ) ) = ( ( 4 x. K ) + 3 ) ) |
55 |
31 43 54
|
3eqtrd |
|- ( K e. NN0 -> ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) = ( ( 4 x. K ) + 3 ) ) |
56 |
|
4ne0 |
|- 4 =/= 0 |
57 |
20 56
|
pm3.2i |
|- ( 4 e. CC /\ 4 =/= 0 ) |
58 |
57
|
a1i |
|- ( K e. NN0 -> ( 4 e. CC /\ 4 =/= 0 ) ) |
59 |
|
divdir |
|- ( ( ( 8 x. K ) e. CC /\ 7 e. CC /\ ( 4 e. CC /\ 4 =/= 0 ) ) -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) ) |
60 |
11 13 58 59
|
syl3anc |
|- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) ) |
61 |
|
8cn |
|- 8 e. CC |
62 |
61
|
a1i |
|- ( K e. NN0 -> 8 e. CC ) |
63 |
|
div23 |
|- ( ( 8 e. CC /\ K e. CC /\ ( 4 e. CC /\ 4 =/= 0 ) ) -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) ) |
64 |
62 24 58 63
|
syl3anc |
|- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( ( 8 / 4 ) x. K ) ) |
65 |
17
|
oveq1i |
|- ( 8 / 4 ) = ( ( 4 x. 2 ) / 4 ) |
66 |
22 20 56
|
divcan3i |
|- ( ( 4 x. 2 ) / 4 ) = 2 |
67 |
65 66
|
eqtri |
|- ( 8 / 4 ) = 2 |
68 |
67
|
a1i |
|- ( K e. NN0 -> ( 8 / 4 ) = 2 ) |
69 |
68
|
oveq1d |
|- ( K e. NN0 -> ( ( 8 / 4 ) x. K ) = ( 2 x. K ) ) |
70 |
64 69
|
eqtrd |
|- ( K e. NN0 -> ( ( 8 x. K ) / 4 ) = ( 2 x. K ) ) |
71 |
70
|
oveq1d |
|- ( K e. NN0 -> ( ( ( 8 x. K ) / 4 ) + ( 7 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) ) |
72 |
60 71
|
eqtrd |
|- ( K e. NN0 -> ( ( ( 8 x. K ) + 7 ) / 4 ) = ( ( 2 x. K ) + ( 7 / 4 ) ) ) |
73 |
72
|
fveq2d |
|- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) ) |
74 |
|
3lt4 |
|- 3 < 4 |
75 |
|
2nn0 |
|- 2 e. NN0 |
76 |
75
|
a1i |
|- ( K e. NN0 -> 2 e. NN0 ) |
77 |
76 9
|
nn0mulcld |
|- ( K e. NN0 -> ( 2 x. K ) e. NN0 ) |
78 |
77
|
nn0zd |
|- ( K e. NN0 -> ( 2 x. K ) e. ZZ ) |
79 |
78
|
peano2zd |
|- ( K e. NN0 -> ( ( 2 x. K ) + 1 ) e. ZZ ) |
80 |
|
3nn0 |
|- 3 e. NN0 |
81 |
80
|
a1i |
|- ( K e. NN0 -> 3 e. NN0 ) |
82 |
|
4nn |
|- 4 e. NN |
83 |
82
|
a1i |
|- ( K e. NN0 -> 4 e. NN ) |
84 |
|
adddivflid |
|- ( ( ( ( 2 x. K ) + 1 ) e. ZZ /\ 3 e. NN0 /\ 4 e. NN ) -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) ) |
85 |
79 81 83 84
|
syl3anc |
|- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) ) |
86 |
23 24
|
mulcld |
|- ( K e. NN0 -> ( 2 x. K ) e. CC ) |
87 |
50
|
a1i |
|- ( K e. NN0 -> 3 e. CC ) |
88 |
56
|
a1i |
|- ( K e. NN0 -> 4 =/= 0 ) |
89 |
87 21 88
|
divcld |
|- ( K e. NN0 -> ( 3 / 4 ) e. CC ) |
90 |
86 14 89
|
addassd |
|- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) ) |
91 |
|
4p3e7 |
|- ( 4 + 3 ) = 7 |
92 |
91
|
eqcomi |
|- 7 = ( 4 + 3 ) |
93 |
92
|
oveq1i |
|- ( 7 / 4 ) = ( ( 4 + 3 ) / 4 ) |
94 |
20 50 20 56
|
divdiri |
|- ( ( 4 + 3 ) / 4 ) = ( ( 4 / 4 ) + ( 3 / 4 ) ) |
95 |
20 56
|
dividi |
|- ( 4 / 4 ) = 1 |
96 |
95
|
oveq1i |
|- ( ( 4 / 4 ) + ( 3 / 4 ) ) = ( 1 + ( 3 / 4 ) ) |
97 |
93 94 96
|
3eqtri |
|- ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) |
98 |
97
|
a1i |
|- ( K e. NN0 -> ( 7 / 4 ) = ( 1 + ( 3 / 4 ) ) ) |
99 |
98
|
eqcomd |
|- ( K e. NN0 -> ( 1 + ( 3 / 4 ) ) = ( 7 / 4 ) ) |
100 |
99
|
oveq2d |
|- ( K e. NN0 -> ( ( 2 x. K ) + ( 1 + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) ) |
101 |
90 100
|
eqtrd |
|- ( K e. NN0 -> ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) = ( ( 2 x. K ) + ( 7 / 4 ) ) ) |
102 |
101
|
fveqeq2d |
|- ( K e. NN0 -> ( ( |_ ` ( ( ( 2 x. K ) + 1 ) + ( 3 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) <-> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) ) |
103 |
85 102
|
bitrd |
|- ( K e. NN0 -> ( 3 < 4 <-> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) ) |
104 |
74 103
|
mpbii |
|- ( K e. NN0 -> ( |_ ` ( ( 2 x. K ) + ( 7 / 4 ) ) ) = ( ( 2 x. K ) + 1 ) ) |
105 |
73 104
|
eqtrd |
|- ( K e. NN0 -> ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) = ( ( 2 x. K ) + 1 ) ) |
106 |
55 105
|
oveq12d |
|- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) ) |
107 |
77
|
nn0cnd |
|- ( K e. NN0 -> ( 2 x. K ) e. CC ) |
108 |
35 87 107 14
|
addsub4d |
|- ( K e. NN0 -> ( ( ( 4 x. K ) + 3 ) - ( ( 2 x. K ) + 1 ) ) = ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) ) |
109 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
110 |
109
|
eqcomi |
|- 4 = ( 2 x. 2 ) |
111 |
110
|
a1i |
|- ( K e. NN0 -> 4 = ( 2 x. 2 ) ) |
112 |
111
|
oveq1d |
|- ( K e. NN0 -> ( 4 x. K ) = ( ( 2 x. 2 ) x. K ) ) |
113 |
23 23 24
|
mulassd |
|- ( K e. NN0 -> ( ( 2 x. 2 ) x. K ) = ( 2 x. ( 2 x. K ) ) ) |
114 |
112 113
|
eqtrd |
|- ( K e. NN0 -> ( 4 x. K ) = ( 2 x. ( 2 x. K ) ) ) |
115 |
114
|
oveq1d |
|- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) ) |
116 |
|
2txmxeqx |
|- ( ( 2 x. K ) e. CC -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) ) |
117 |
107 116
|
syl |
|- ( K e. NN0 -> ( ( 2 x. ( 2 x. K ) ) - ( 2 x. K ) ) = ( 2 x. K ) ) |
118 |
115 117
|
eqtrd |
|- ( K e. NN0 -> ( ( 4 x. K ) - ( 2 x. K ) ) = ( 2 x. K ) ) |
119 |
|
3m1e2 |
|- ( 3 - 1 ) = 2 |
120 |
119
|
a1i |
|- ( K e. NN0 -> ( 3 - 1 ) = 2 ) |
121 |
118 120
|
oveq12d |
|- ( K e. NN0 -> ( ( ( 4 x. K ) - ( 2 x. K ) ) + ( 3 - 1 ) ) = ( ( 2 x. K ) + 2 ) ) |
122 |
106 108 121
|
3eqtrd |
|- ( K e. NN0 -> ( ( ( ( ( 8 x. K ) + 7 ) - 1 ) / 2 ) - ( |_ ` ( ( ( 8 x. K ) + 7 ) / 4 ) ) ) = ( ( 2 x. K ) + 2 ) ) |
123 |
6 122
|
sylan9eqr |
|- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) ) |