Step |
Hyp |
Ref |
Expression |
1 |
|
qnumdencoprm |
|- ( A e. QQ -> ( ( numer ` A ) gcd ( denom ` A ) ) = 1 ) |
2 |
1
|
adantr |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( numer ` A ) gcd ( denom ` A ) ) = 1 ) |
3 |
2
|
oveq1d |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( ( numer ` A ) gcd ( denom ` A ) ) ^ N ) = ( 1 ^ N ) ) |
4 |
|
qnumcl |
|- ( A e. QQ -> ( numer ` A ) e. ZZ ) |
5 |
4
|
adantr |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( numer ` A ) e. ZZ ) |
6 |
|
qdencl |
|- ( A e. QQ -> ( denom ` A ) e. NN ) |
7 |
6
|
adantr |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( denom ` A ) e. NN ) |
8 |
7
|
nnzd |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( denom ` A ) e. ZZ ) |
9 |
|
simpr |
|- ( ( A e. QQ /\ N e. NN0 ) -> N e. NN0 ) |
10 |
|
zexpgcd |
|- ( ( ( numer ` A ) e. ZZ /\ ( denom ` A ) e. ZZ /\ N e. NN0 ) -> ( ( ( numer ` A ) gcd ( denom ` A ) ) ^ N ) = ( ( ( numer ` A ) ^ N ) gcd ( ( denom ` A ) ^ N ) ) ) |
11 |
5 8 9 10
|
syl3anc |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( ( numer ` A ) gcd ( denom ` A ) ) ^ N ) = ( ( ( numer ` A ) ^ N ) gcd ( ( denom ` A ) ^ N ) ) ) |
12 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
13 |
|
1exp |
|- ( N e. ZZ -> ( 1 ^ N ) = 1 ) |
14 |
9 12 13
|
3syl |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( 1 ^ N ) = 1 ) |
15 |
3 11 14
|
3eqtr3d |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( ( numer ` A ) ^ N ) gcd ( ( denom ` A ) ^ N ) ) = 1 ) |
16 |
|
qeqnumdivden |
|- ( A e. QQ -> A = ( ( numer ` A ) / ( denom ` A ) ) ) |
17 |
16
|
adantr |
|- ( ( A e. QQ /\ N e. NN0 ) -> A = ( ( numer ` A ) / ( denom ` A ) ) ) |
18 |
17
|
oveq1d |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) = ( ( ( numer ` A ) / ( denom ` A ) ) ^ N ) ) |
19 |
5
|
zcnd |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( numer ` A ) e. CC ) |
20 |
7
|
nncnd |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( denom ` A ) e. CC ) |
21 |
7
|
nnne0d |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( denom ` A ) =/= 0 ) |
22 |
19 20 21 9
|
expdivd |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( ( numer ` A ) / ( denom ` A ) ) ^ N ) = ( ( ( numer ` A ) ^ N ) / ( ( denom ` A ) ^ N ) ) ) |
23 |
18 22
|
eqtrd |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) = ( ( ( numer ` A ) ^ N ) / ( ( denom ` A ) ^ N ) ) ) |
24 |
|
qexpcl |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ ) |
25 |
|
zexpcl |
|- ( ( ( numer ` A ) e. ZZ /\ N e. NN0 ) -> ( ( numer ` A ) ^ N ) e. ZZ ) |
26 |
4 25
|
sylan |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( numer ` A ) ^ N ) e. ZZ ) |
27 |
7 9
|
nnexpcld |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( denom ` A ) ^ N ) e. NN ) |
28 |
|
qnumdenbi |
|- ( ( ( A ^ N ) e. QQ /\ ( ( numer ` A ) ^ N ) e. ZZ /\ ( ( denom ` A ) ^ N ) e. NN ) -> ( ( ( ( ( numer ` A ) ^ N ) gcd ( ( denom ` A ) ^ N ) ) = 1 /\ ( A ^ N ) = ( ( ( numer ` A ) ^ N ) / ( ( denom ` A ) ^ N ) ) ) <-> ( ( numer ` ( A ^ N ) ) = ( ( numer ` A ) ^ N ) /\ ( denom ` ( A ^ N ) ) = ( ( denom ` A ) ^ N ) ) ) ) |
29 |
24 26 27 28
|
syl3anc |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( ( ( ( numer ` A ) ^ N ) gcd ( ( denom ` A ) ^ N ) ) = 1 /\ ( A ^ N ) = ( ( ( numer ` A ) ^ N ) / ( ( denom ` A ) ^ N ) ) ) <-> ( ( numer ` ( A ^ N ) ) = ( ( numer ` A ) ^ N ) /\ ( denom ` ( A ^ N ) ) = ( ( denom ` A ) ^ N ) ) ) ) |
30 |
15 23 29
|
mpbi2and |
|- ( ( A e. QQ /\ N e. NN0 ) -> ( ( numer ` ( A ^ N ) ) = ( ( numer ` A ) ^ N ) /\ ( denom ` ( A ^ N ) ) = ( ( denom ` A ) ^ N ) ) ) |