Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> A e. ZZ ) |
2 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
3 |
2
|
3ad2ant2 |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> N e. ZZ ) |
4 |
|
nnne0 |
|- ( N e. NN -> N =/= 0 ) |
5 |
4
|
3ad2ant2 |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> N =/= 0 ) |
6 |
|
lgsdi |
|- ( ( ( A e. ZZ /\ N e. ZZ /\ N e. ZZ ) /\ ( N =/= 0 /\ N =/= 0 ) ) -> ( A /L ( N x. N ) ) = ( ( A /L N ) x. ( A /L N ) ) ) |
7 |
1 3 3 5 5 6
|
syl32anc |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L ( N x. N ) ) = ( ( A /L N ) x. ( A /L N ) ) ) |
8 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
9 |
8
|
3ad2ant2 |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> N e. CC ) |
10 |
9
|
sqvald |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( N ^ 2 ) = ( N x. N ) ) |
11 |
10
|
oveq2d |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L ( N ^ 2 ) ) = ( A /L ( N x. N ) ) ) |
12 |
|
lgscl |
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ ) |
13 |
1 3 12
|
syl2anc |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. ZZ ) |
14 |
13
|
zred |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR ) |
15 |
|
absresq |
|- ( ( A /L N ) e. RR -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) ) |
16 |
14 15
|
syl |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) ) |
17 |
|
lgsabs1 |
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) ) |
18 |
2 17
|
sylan2 |
|- ( ( A e. ZZ /\ N e. NN ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) ) |
19 |
18
|
biimp3ar |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 ) |
20 |
19
|
oveq1d |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) ) |
21 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
22 |
20 21
|
eqtrdi |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 ) |
23 |
13
|
zcnd |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC ) |
24 |
23
|
sqvald |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( ( A /L N ) ^ 2 ) = ( ( A /L N ) x. ( A /L N ) ) ) |
25 |
16 22 24
|
3eqtr3d |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> 1 = ( ( A /L N ) x. ( A /L N ) ) ) |
26 |
7 11 25
|
3eqtr4d |
|- ( ( A e. ZZ /\ N e. NN /\ ( A gcd N ) = 1 ) -> ( A /L ( N ^ 2 ) ) = 1 ) |