Step |
Hyp |
Ref |
Expression |
1 |
|
2lgslem2.n |
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) |
2 |
|
nnnn0 |
|- ( P e. NN -> P e. NN0 ) |
3 |
|
8nn |
|- 8 e. NN |
4 |
|
nnrp |
|- ( 8 e. NN -> 8 e. RR+ ) |
5 |
3 4
|
ax-mp |
|- 8 e. RR+ |
6 |
|
modmuladdnn0 |
|- ( ( P e. NN0 /\ 8 e. RR+ ) -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) ) |
7 |
2 5 6
|
sylancl |
|- ( P e. NN -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) ) |
8 |
|
simpr |
|- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 ) |
9 |
|
nn0cn |
|- ( k e. NN0 -> k e. CC ) |
10 |
|
8cn |
|- 8 e. CC |
11 |
10
|
a1i |
|- ( k e. NN0 -> 8 e. CC ) |
12 |
9 11
|
mulcomd |
|- ( k e. NN0 -> ( k x. 8 ) = ( 8 x. k ) ) |
13 |
12
|
adantl |
|- ( ( P e. NN /\ k e. NN0 ) -> ( k x. 8 ) = ( 8 x. k ) ) |
14 |
13
|
oveq1d |
|- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) ) |
15 |
14
|
eqeq2d |
|- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 7 ) <-> P = ( ( 8 x. k ) + 7 ) ) ) |
16 |
15
|
biimpa |
|- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> P = ( ( 8 x. k ) + 7 ) ) |
17 |
1
|
2lgslem3d |
|- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) ) |
18 |
8 16 17
|
syl2an2r |
|- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) ) |
19 |
|
oveq1 |
|- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) ) |
20 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
21 |
20
|
eqcomi |
|- 2 = ( 2 x. 1 ) |
22 |
21
|
a1i |
|- ( k e. NN0 -> 2 = ( 2 x. 1 ) ) |
23 |
22
|
oveq2d |
|- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) ) |
24 |
|
2cnd |
|- ( k e. NN0 -> 2 e. CC ) |
25 |
|
1cnd |
|- ( k e. NN0 -> 1 e. CC ) |
26 |
|
adddi |
|- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) ) |
27 |
26
|
eqcomd |
|- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) ) |
28 |
24 9 25 27
|
syl3anc |
|- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) ) |
29 |
9 25
|
addcld |
|- ( k e. NN0 -> ( k + 1 ) e. CC ) |
30 |
24 29
|
mulcomd |
|- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) ) |
31 |
23 28 30
|
3eqtrd |
|- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) ) |
32 |
31
|
oveq1d |
|- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) ) |
33 |
|
peano2nn0 |
|- ( k e. NN0 -> ( k + 1 ) e. NN0 ) |
34 |
33
|
nn0zd |
|- ( k e. NN0 -> ( k + 1 ) e. ZZ ) |
35 |
|
2rp |
|- 2 e. RR+ |
36 |
|
mulmod0 |
|- ( ( ( k + 1 ) e. ZZ /\ 2 e. RR+ ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 ) |
37 |
34 35 36
|
sylancl |
|- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 ) |
38 |
32 37
|
eqtrd |
|- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 ) |
39 |
19 38
|
sylan9eqr |
|- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 2 ) ) -> ( N mod 2 ) = 0 ) |
40 |
8 18 39
|
syl2an2r |
|- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> ( N mod 2 ) = 0 ) |
41 |
40
|
rexlimdva2 |
|- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 7 ) -> ( N mod 2 ) = 0 ) ) |
42 |
7 41
|
syld |
|- ( P e. NN -> ( ( P mod 8 ) = 7 -> ( N mod 2 ) = 0 ) ) |
43 |
42
|
imp |
|- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 ) |