Step |
Hyp |
Ref |
Expression |
1 |
|
simp1l |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. ZZ ) |
2 |
|
simp2 |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> N e. ZZ ) |
3 |
|
simp1r |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A =/= 0 ) |
4 |
|
lgsdir |
|- ( ( ( A e. ZZ /\ A e. ZZ /\ N e. ZZ ) /\ ( A =/= 0 /\ A =/= 0 ) ) -> ( ( A x. A ) /L N ) = ( ( A /L N ) x. ( A /L N ) ) ) |
5 |
1 1 2 3 3 4
|
syl32anc |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A x. A ) /L N ) = ( ( A /L N ) x. ( A /L N ) ) ) |
6 |
|
zcn |
|- ( A e. ZZ -> A e. CC ) |
7 |
6
|
adantr |
|- ( ( A e. ZZ /\ A =/= 0 ) -> A e. CC ) |
8 |
7
|
3ad2ant1 |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> A e. CC ) |
9 |
8
|
sqvald |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A ^ 2 ) = ( A x. A ) ) |
10 |
9
|
oveq1d |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = ( ( A x. A ) /L N ) ) |
11 |
|
lgscl |
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ ) |
12 |
1 2 11
|
syl2anc |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. ZZ ) |
13 |
12
|
zred |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. RR ) |
14 |
|
absresq |
|- ( ( A /L N ) e. RR -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) ) |
15 |
13 14
|
syl |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( ( A /L N ) ^ 2 ) ) |
16 |
|
lgsabs1 |
|- ( ( A e. ZZ /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) ) |
17 |
16
|
adantlr |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ ) -> ( ( abs ` ( A /L N ) ) = 1 <-> ( A gcd N ) = 1 ) ) |
18 |
17
|
biimp3ar |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( abs ` ( A /L N ) ) = 1 ) |
19 |
18
|
oveq1d |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = ( 1 ^ 2 ) ) |
20 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
21 |
19 20
|
eqtrdi |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( abs ` ( A /L N ) ) ^ 2 ) = 1 ) |
22 |
12
|
zcnd |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( A /L N ) e. CC ) |
23 |
22
|
sqvald |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A /L N ) ^ 2 ) = ( ( A /L N ) x. ( A /L N ) ) ) |
24 |
15 21 23
|
3eqtr3d |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> 1 = ( ( A /L N ) x. ( A /L N ) ) ) |
25 |
5 10 24
|
3eqtr4d |
|- ( ( ( A e. ZZ /\ A =/= 0 ) /\ N e. ZZ /\ ( A gcd N ) = 1 ) -> ( ( A ^ 2 ) /L N ) = 1 ) |