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Theorem rpexp 10739
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 9660 . . . . . 6  |-  ( N  e.  NN  ->  (
0 ^ N )  =  0 )
21oveq1d 5578 . . . . 5  |-  ( N  e.  NN  ->  (
( 0 ^ N
)  gcd  0 )  =  ( 0  gcd  0 ) )
32eqeq1d 2091 . . . 4  |-  ( N  e.  NN  ->  (
( ( 0 ^ N )  gcd  0
)  =  1  <->  (
0  gcd  0 )  =  1 ) )
4 oveq1 5570 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
5 oveq12 5572 . . . . . . 7  |-  ( ( ( A ^ N
)  =  ( 0 ^ N )  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
64, 5sylan 277 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
76eqeq1d 2091 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( ( 0 ^ N )  gcd  0 )  =  1 ) )
8 oveq12 5572 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
98eqeq1d 2091 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B )  =  1  <->  ( 0  gcd  0 )  =  1 ) )
107, 9bibi12d 233 . . . 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 155 . . 3  |-  ( N  e.  NN  ->  (
( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
12113ad2ant3 962 . 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 10726 . . . . . . 7  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( A ^ N )  gcd 
B ) )
14 simpl1 942 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
15 simpl3 944 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN )
1615nnnn0d 8460 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN0 )
17 zexpcl 9640 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  ZZ )
1814, 16, 17syl2anc 403 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A ^ N
)  e.  ZZ )
1918adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
20 simpl2 943 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  B  e.  ZZ )
2120adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  B  e.  ZZ )
22 gcddvds 10562 . . . . . . . . . . . . . . 15  |-  ( ( ( A ^ N
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( A ^ N )  gcd 
B )  ||  ( A ^ N )  /\  ( ( A ^ N )  gcd  B
)  ||  B )
)
2319, 21, 22syl2anc 403 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 10700 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2625adantl 271 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
27 simpr 108 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
2814zcnd 8603 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  CC )
29 expeq0 9656 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3028, 15, 29syl2anc 403 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3130anbi1d 453 . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )
33 gcdn0cl 10561 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )  ->  ( ( A ^ N )  gcd 
B )  e.  NN )
3418, 20, 32, 33syl21anc 1169 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  NN )
3534nnzd 8601 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  ZZ )
3635adantr 270 . . . . . . . . . . . . . 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 10440 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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 108 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
41 simpll1 978 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  e.  ZZ )
4215adantr 270 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  N  e.  NN )
43 prmdvdsexp 10734 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4440, 41, 42, 43syl3anc 1170 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4539, 44sylibd 147 . . . . . . . . . . 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 112 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A ^ N
)  gcd  B )  ||  B )
47 dvdstr 10440 . . . . . . . . . . . . 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 1170 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 301 . . . . . . . . . 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 10608 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
5226, 41, 21, 51syl3anc 1170 . . . . . . . . . 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 10728 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
54 breq2 3809 . . . . . . . . . . . . . 14  |-  ( ( A  gcd  B )  =  1  ->  (
p  ||  ( A  gcd  B )  <->  p  ||  1
) )
5554notbid 625 . . . . . . . . . . . . 13  |-  ( ( A  gcd  B )  =  1  ->  ( -.  p  ||  ( A  gcd  B )  <->  -.  p  ||  1 ) )
5653, 55syl5ibrcom 155 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( A  gcd  B )  =  1  ->  -.  p  ||  ( A  gcd  B ) ) )
5756necon2ad 2306 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( A  gcd  B )  ->  ( A  gcd  B )  =/=  1
) )
5857adantl 271 . . . . . . . . . 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 56 . . . . . . . . 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 2482 . . . . . . . 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 10561 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
62613adantl3 1097 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  NN )
63 eluz2b3 8824 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
6463baib 862 . . . . . . . . 9  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  e.  ( ZZ>= ` 
2 )  <->  ( A  gcd  B )  =/=  1
) )
6562, 64syl 14 . . . . . . . 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 167 . . . . . . 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 32 . . . . . 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 10726 . . . . . . 7  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( A  gcd  B ) )
69 gcddvds 10562 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
7041, 21, 69syl2anc 403 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  A )
72 iddvdsexp 10427 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  A  ||  ( A ^ N ) )
7341, 42, 72syl2anc 403 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  ||  ( A ^ N
) )
7462nnzd 8601 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
7574adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  e.  ZZ )
76 dvdstr 10440 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  ( A ^ N ) )
79 dvdstr 10440 . . . . . . . . . . . . 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 1170 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 112 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  B )
83 dvdstr 10440 . . . . . . . . . . . . 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 1170 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 301 . . . . . . . . . 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 10608 . . . . . . . . . . 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 1170 . . . . . . . . . 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 3809 . . . . . . . . . . . . . 14  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  <->  p  ||  1
) )
9089notbid 625 . . . . . . . . . . . . 13  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  ( -.  p  ||  ( ( A ^ N )  gcd  B )  <->  -.  p  ||  1 ) )
9153, 90syl5ibrcom 155 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( ( A ^ N
)  gcd  B )  =  1  ->  -.  p  ||  ( ( A ^ N )  gcd 
B ) ) )
9291necon2ad 2306 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( ( A ^ N )  gcd 
B )  ->  (
( A ^ N
)  gcd  B )  =/=  1 ) )
9392adantl 271 . . . . . . . . . 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 56 . . . . . . . . 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 2482 . . . . . . . 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 8824 . . . . . . . . . 10  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( ( A ^ N )  gcd  B
)  e.  NN  /\  ( ( A ^ N )  gcd  B
)  =/=  1 ) )
9796baib 862 . . . . . . . . 9  |-  ( ( ( A ^ N
)  gcd  B )  e.  NN  ->  ( (
( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( A ^ N
)  gcd  B )  =/=  1 ) )
9834, 97syl 14 . . . . . . . 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 167 . . . . . . 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 32 . . . . . 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 127 . . . . 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 216 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) )
103 simp1 939 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  A  e.  ZZ )
104 simp3 941 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  N  e.  NN )
105104nnnn0d 8460 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  N  e.  NN0 )
106103, 105, 17syl2anc 403 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A ^ N )  e.  ZZ )
107 simp2 940 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  B  e.  ZZ )
108106, 107gcdcld 10567 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A ^ N
)  gcd  B )  e.  NN0 )
109108nn0zd 8600 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A ^ N
)  gcd  B )  e.  ZZ )
110 1zzd 8511 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  1  e.  ZZ )
111 zdceq 8556 . . . . . . 7  |-  ( ( ( ( A ^ N )  gcd  B
)  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( ( A ^ N
)  gcd  B )  =  1 )
112109, 110, 111syl2anc 403 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  -> DECID  ( ( A ^ N )  gcd  B
)  =  1 )
113103, 107gcdcld 10567 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  gcd  B )  e. 
NN0 )
114113nn0zd 8600 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  gcd  B )  e.  ZZ )
115 zdceq 8556 . . . . . . 7  |-  ( ( ( A  gcd  B
)  e.  ZZ  /\  1  e.  ZZ )  -> DECID  ( A  gcd  B )  =  1 )
116114, 110, 115syl2anc 403 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  -> DECID  ( A  gcd  B
)  =  1 )
117 nebidc 2329 . . . . . 6  |-  (DECID  ( ( A ^ N )  gcd  B )  =  1  ->  (DECID  ( A  gcd  B )  =  1  ->  ( ( ( ( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 )  <->  ( ( ( A ^ N )  gcd  B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) ) ) )
118112, 116, 117sylc 61 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( ( ( A ^ N )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 )  <-> 
( ( ( A ^ N )  gcd 
B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) ) )
119118adantr 270 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( ( A ^ N )  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 )  <-> 
( ( ( A ^ N )  gcd 
B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) ) )
120102, 119mpbird 165 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
121120ex 113 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( -.  ( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
122 gcdmndc 10547 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  ( A  =  0  /\  B  =  0 ) )
123 exmiddc 778 . . . 4  |-  (DECID  ( A  =  0  /\  B  =  0 )  -> 
( ( A  =  0  /\  B  =  0 )  \/  -.  ( A  =  0  /\  B  =  0
) ) )
124122, 123syl 14 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  =  0  /\  B  =  0 )  \/  -.  ( A  =  0  /\  B  =  0
) ) )
1251243adant3 959 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A  =  0  /\  B  =  0 )  \/  -.  ( A  =  0  /\  B  =  0 ) ) )
12612, 121, 125mpjaod 671 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    /\ wa 102    <-> wb 103    \/ wo 662  DECID wdc 776    /\ w3a 920    = wceq 1285    e. wcel 1434    =/= wne 2249   E.wrex 2354   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096   NNcn 8158   2c2 8208   NN0cn0 8407   ZZcz 8484   ZZ>=cuz 8752   ^cexp 9624    || cdvds 10403    gcd cgcd 10545   Primecprime 10696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-1o 6085  df-2o 6086  df-er 6193  df-en 6309  df-sup 6491  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-gcd 10546  df-prm 10697
This theorem is referenced by:  rpexp1i  10740  phiprmpw  10805
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