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Theorem rpexp 12799
Description: If two numbers  A and  B are relatively prime, then they are still relatively prime if raised to a power. (Contributed by Mario Carneiro, 24-Feb-2014.)
Assertion
Ref Expression
rpexp  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( ( A ^ N )  gcd  B
)  =  1  <->  ( A  gcd  B )  =  1 ) )

Proof of Theorem rpexp
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 0exp 11137 . . . . . 6  |-  ( N  e.  NN  ->  (
0 ^ N )  =  0 )
21oveq1d 5873 . . . . 5  |-  ( N  e.  NN  ->  (
( 0 ^ N
)  gcd  0 )  =  ( 0  gcd  0 ) )
32eqeq1d 2291 . . . 4  |-  ( N  e.  NN  ->  (
( ( 0 ^ N )  gcd  0
)  =  1  <->  (
0  gcd  0 )  =  1 ) )
4 oveq1 5865 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
5 oveq12 5867 . . . . . . 7  |-  ( ( ( A ^ N
)  =  ( 0 ^ N )  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
64, 5sylan 457 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
76eqeq1d 2291 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( ( 0 ^ N )  gcd  0 )  =  1 ) )
8 oveq12 5867 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
98eqeq1d 2291 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B )  =  1  <->  ( 0  gcd  0 )  =  1 ) )
107, 9bibi12d 312 . . . 4  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( ( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 )  <->  ( ( ( 0 ^ N )  gcd  0 )  =  1  <->  ( 0  gcd  0 )  =  1 ) ) )
113, 10syl5ibrcom 213 . . 3  |-  ( N  e.  NN  ->  (
( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
12113ad2ant3 978 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
13 exprmfct 12789 . . . . . . 7  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( A ^ N )  gcd 
B ) )
14 simpl1 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
15 simpl3 960 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN )
1615nnnn0d 10018 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN0 )
17 zexpcl 11118 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  ZZ )
1814, 16, 17syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A ^ N
)  e.  ZZ )
1918adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
20 simpl2 959 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  B  e.  ZZ )
2120adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  B  e.  ZZ )
22 gcddvds 12694 . . . . . . . . . . . . . . 15  |-  ( ( ( A ^ N
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( A ^ N )  gcd 
B )  ||  ( A ^ N )  /\  ( ( A ^ N )  gcd  B
)  ||  B )
)
2319, 21, 22syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( ( A ^ N )  gcd  B
)  ||  ( A ^ N )  /\  (
( A ^ N
)  gcd  B )  ||  B ) )
2423simpld 445 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A ^ N
)  gcd  B )  ||  ( A ^ N
) )
25 prmz 12762 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2625adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
27 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
2814zcnd 10118 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  CC )
29 expeq0 11132 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3028, 15, 29syl2anc 642 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3130anbi1d 685 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  =  0  /\  B  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
3227, 31mtbird 292 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )
33 gcdn0cl 12693 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )  ->  ( ( A ^ N )  gcd 
B )  e.  NN )
3418, 20, 32, 33syl21anc 1181 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  NN )
3534nnzd 10116 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  ZZ )
3635adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A ^ N
)  gcd  B )  e.  ZZ )
37 dvdstr 12562 . . . . . . . . . . . . . 14  |-  ( ( p  e.  ZZ  /\  ( ( A ^ N )  gcd  B
)  e.  ZZ  /\  ( A ^ N )  e.  ZZ )  -> 
( ( p  ||  ( ( A ^ N )  gcd  B
)  /\  ( ( A ^ N )  gcd 
B )  ||  ( A ^ N ) )  ->  p  ||  ( A ^ N ) ) )
3826, 36, 19, 37syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  (
( A ^ N
)  gcd  B )  /\  ( ( A ^ N )  gcd  B
)  ||  ( A ^ N ) )  ->  p  ||  ( A ^ N ) ) )
3924, 38mpan2d 655 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  p  ||  ( A ^ N
) ) )
40 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
41 simpll1 994 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  e.  ZZ )
4215adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  N  e.  NN )
43 prmdvdsexp 12793 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4440, 41, 42, 43syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4539, 44sylibd 205 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  p  ||  A ) )
4623simprd 449 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A ^ N
)  gcd  B )  ||  B )
47 dvdstr 12562 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  ( ( A ^ N )  gcd  B
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( p  ||  ( ( A ^ N )  gcd  B
)  /\  ( ( A ^ N )  gcd 
B )  ||  B
)  ->  p  ||  B
) )
4826, 36, 21, 47syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  (
( A ^ N
)  gcd  B )  /\  ( ( A ^ N )  gcd  B
)  ||  B )  ->  p  ||  B ) )
4946, 48mpan2d 655 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  p  ||  B ) )
5045, 49jcad 519 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  (
p  ||  A  /\  p  ||  B ) ) )
51 dvdsgcd 12722 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
5226, 41, 21, 51syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
53 nprmdvds1 12790 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
54 breq2 4027 . . . . . . . . . . . . . 14  |-  ( ( A  gcd  B )  =  1  ->  (
p  ||  ( A  gcd  B )  <->  p  ||  1
) )
5554notbid 285 . . . . . . . . . . . . 13  |-  ( ( A  gcd  B )  =  1  ->  ( -.  p  ||  ( A  gcd  B )  <->  -.  p  ||  1 ) )
5653, 55syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( A  gcd  B )  =  1  ->  -.  p  ||  ( A  gcd  B ) ) )
5756necon2ad 2494 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( A  gcd  B )  ->  ( A  gcd  B )  =/=  1
) )
5857adantl 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A  gcd  B )  ->  ( A  gcd  B )  =/=  1 ) )
5950, 52, 583syld 51 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  ( A  gcd  B )  =/=  1 ) )
6059rexlimdva 2667 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( E. p  e. 
Prime  p  ||  ( ( A ^ N )  gcd  B )  -> 
( A  gcd  B
)  =/=  1 ) )
61 gcdn0cl 12693 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
62613adantl3 1113 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  NN )
63 eluz2b3 10291 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
6463baib 871 . . . . . . . . 9  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  e.  ( ZZ>= ` 
2 )  <->  ( A  gcd  B )  =/=  1
) )
6562, 64syl 15 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  gcd  B )  e.  ( ZZ>= ` 
2 )  <->  ( A  gcd  B )  =/=  1
) )
6660, 65sylibrd 225 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( E. p  e. 
Prime  p  ||  ( ( A ^ N )  gcd  B )  -> 
( A  gcd  B
)  e.  ( ZZ>= ` 
2 ) ) )
6713, 66syl5 28 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  e.  (
ZZ>= `  2 )  -> 
( A  gcd  B
)  e.  ( ZZ>= ` 
2 ) ) )
68 exprmfct 12789 . . . . . . 7  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( A  gcd  B ) )
69 gcddvds 12694 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
7041, 21, 69syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A  gcd  B
)  ||  A  /\  ( A  gcd  B ) 
||  B ) )
7170simpld 445 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  A )
72 iddvdsexp 12552 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  A  ||  ( A ^ N ) )
7341, 42, 72syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  ||  ( A ^ N
) )
7462nnzd 10116 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
7574adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  e.  ZZ )
76 dvdstr 12562 . . . . . . . . . . . . . 14  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  ( A ^ N )  e.  ZZ )  ->  (
( ( A  gcd  B )  ||  A  /\  A  ||  ( A ^ N ) )  -> 
( A  gcd  B
)  ||  ( A ^ N ) ) )
7775, 41, 19, 76syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( ( A  gcd  B )  ||  A  /\  A  ||  ( A ^ N ) )  -> 
( A  gcd  B
)  ||  ( A ^ N ) ) )
7871, 73, 77mp2and 660 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  ( A ^ N ) )
79 dvdstr 12562 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  ( A  gcd  B )  e.  ZZ  /\  ( A ^ N )  e.  ZZ )  ->  (
( p  ||  ( A  gcd  B )  /\  ( A  gcd  B ) 
||  ( A ^ N ) )  ->  p  ||  ( A ^ N ) ) )
8026, 75, 19, 79syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  ( A  gcd  B )  /\  ( A  gcd  B ) 
||  ( A ^ N ) )  ->  p  ||  ( A ^ N ) ) )
8178, 80mpan2d 655 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A  gcd  B )  ->  p  ||  ( A ^ N
) ) )
8270simprd 449 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  B )
83 dvdstr 12562 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  ( A  gcd  B )  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  ( A  gcd  B )  /\  ( A  gcd  B ) 
||  B )  ->  p  ||  B ) )
8426, 75, 21, 83syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  ( A  gcd  B )  /\  ( A  gcd  B ) 
||  B )  ->  p  ||  B ) )
8582, 84mpan2d 655 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A  gcd  B )  ->  p  ||  B ) )
8681, 85jcad 519 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A  gcd  B )  ->  (
p  ||  ( A ^ N )  /\  p  ||  B ) ) )
87 dvdsgcd 12722 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  ( A ^ N )  /\  p  ||  B )  ->  p  ||  ( ( A ^ N )  gcd 
B ) ) )
8826, 19, 21, 87syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( p  ||  ( A ^ N )  /\  p  ||  B )  ->  p  ||  ( ( A ^ N )  gcd 
B ) ) )
89 breq2 4027 . . . . . . . . . . . . . 14  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  <->  p  ||  1
) )
9089notbid 285 . . . . . . . . . . . . 13  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  ( -.  p  ||  ( ( A ^ N )  gcd  B )  <->  -.  p  ||  1 ) )
9153, 90syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( ( A ^ N
)  gcd  B )  =  1  ->  -.  p  ||  ( ( A ^ N )  gcd 
B ) ) )
9291necon2ad 2494 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( ( A ^ N )  gcd 
B )  ->  (
( A ^ N
)  gcd  B )  =/=  1 ) )
9392adantl 452 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  ->  (
( A ^ N
)  gcd  B )  =/=  1 ) )
9486, 88, 933syld 51 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A  gcd  B )  ->  (
( A ^ N
)  gcd  B )  =/=  1 ) )
9594rexlimdva 2667 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( E. p  e. 
Prime  p  ||  ( A  gcd  B )  -> 
( ( A ^ N )  gcd  B
)  =/=  1 ) )
96 eluz2b3 10291 . . . . . . . . . 10  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( ( A ^ N )  gcd  B
)  e.  NN  /\  ( ( A ^ N )  gcd  B
)  =/=  1 ) )
9796baib 871 . . . . . . . . 9  |-  ( ( ( A ^ N
)  gcd  B )  e.  NN  ->  ( (
( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( A ^ N
)  gcd  B )  =/=  1 ) )
9834, 97syl 15 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  e.  (
ZZ>= `  2 )  <->  ( ( A ^ N )  gcd 
B )  =/=  1
) )
9995, 98sylibrd 225 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( E. p  e. 
Prime  p  ||  ( A  gcd  B )  -> 
( ( A ^ N )  gcd  B
)  e.  ( ZZ>= ` 
2 ) ) )
10068, 99syl5 28 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A  gcd  B )  e.  ( ZZ>= ` 
2 )  ->  (
( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )
) )
10167, 100impbid 183 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  e.  (
ZZ>= `  2 )  <->  ( A  gcd  B )  e.  (
ZZ>= `  2 ) ) )
102101, 98, 653bitr3d 274 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) )
103102necon4bid 2512 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
104103ex 423 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( -.  ( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
10512, 104pm2.61d 150 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( ( A ^ N )  gcd  B
)  =  1  <->  ( A  gcd  B )  =  1 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758
This theorem is referenced by:  rpexp1i  12800  phiprmpw  12844  pockthlem  12952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759
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