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Theorem rpexp 14113
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 12163 . . . . . 6  |-  ( N  e.  NN  ->  (
0 ^ N )  =  0 )
21oveq1d 6297 . . . . 5  |-  ( N  e.  NN  ->  (
( 0 ^ N
)  gcd  0 )  =  ( 0  gcd  0 ) )
32eqeq1d 2469 . . . 4  |-  ( N  e.  NN  ->  (
( ( 0 ^ N )  gcd  0
)  =  1  <->  (
0  gcd  0 )  =  1 ) )
4 oveq1 6289 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
5 oveq12 6291 . . . . . . 7  |-  ( ( ( A ^ N
)  =  ( 0 ^ N )  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
64, 5sylan 471 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A ^ N )  gcd 
B )  =  ( ( 0 ^ N
)  gcd  0 ) )
76eqeq1d 2469 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( ( 0 ^ N )  gcd  0 )  =  1 ) )
8 oveq12 6291 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
98eqeq1d 2469 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( A  gcd  B )  =  1  <->  ( 0  gcd  0 )  =  1 ) )
107, 9bibi12d 321 . . . 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 222 . . 3  |-  ( N  e.  NN  ->  (
( A  =  0  /\  B  =  0 )  ->  ( (
( A ^ N
)  gcd  B )  =  1  <->  ( A  gcd  B )  =  1 ) ) )
12113ad2ant3 1019 . 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 14103 . . . . . . 7  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( A ^ N )  gcd 
B ) )
14 simpl1 999 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
15 simpl3 1001 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN )
1615nnnn0d 10848 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN0 )
17 zexpcl 12144 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A ^ N
)  e.  ZZ )
1814, 16, 17syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A ^ N
)  e.  ZZ )
1918adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A ^ N )  e.  ZZ )
20 simpl2 1000 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  B  e.  ZZ )
2120adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  B  e.  ZZ )
22 gcddvds 14005 . . . . . . . . . . . . . . 15  |-  ( ( ( A ^ N
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( A ^ N )  gcd 
B )  ||  ( A ^ N )  /\  ( ( A ^ N )  gcd  B
)  ||  B )
)
2319, 21, 22syl2anc 661 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 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 14073 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
2625adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  ZZ )
27 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
2814zcnd 10963 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  CC )
29 expeq0 12158 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A  e.  CC  /\  N  e.  NN )  ->  ( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3028, 15, 29syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  =  0  <-> 
A  =  0 ) )
3130anbi1d 704 . . . . . . . . . . . . . . . . . 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 301 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )
33 gcdn0cl 14004 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )  ->  ( ( A ^ N )  gcd 
B )  e.  NN )
3418, 20, 32, 33syl21anc 1227 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  NN )
3534nnzd 10961 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  ZZ )
3635adantr 465 . . . . . . . . . . . . . 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 13871 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 674 . . . . . . . . . . . 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 461 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
41 simpll1 1035 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  e.  ZZ )
4215adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  N  e.  NN )
43 prmdvdsexp 14107 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4440, 41, 42, 43syl3anc 1228 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4539, 44sylibd 214 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  (
( A ^ N
)  gcd  B )  ||  B )
47 dvdstr 13871 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 674 . . . . . . . . . . 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 533 . . . . . . . . . 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 14033 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
5226, 41, 21, 51syl3anc 1228 . . . . . . . . . 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 14104 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
54 breq2 4451 . . . . . . . . . . . . . 14  |-  ( ( A  gcd  B )  =  1  ->  (
p  ||  ( A  gcd  B )  <->  p  ||  1
) )
5554notbid 294 . . . . . . . . . . . . 13  |-  ( ( A  gcd  B )  =  1  ->  ( -.  p  ||  ( A  gcd  B )  <->  -.  p  ||  1 ) )
5653, 55syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( A  gcd  B )  =  1  ->  -.  p  ||  ( A  gcd  B ) ) )
5756necon2ad 2680 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( A  gcd  B )  ->  ( A  gcd  B )  =/=  1
) )
5857adantl 466 . . . . . . . . . 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 55 . . . . . . . . 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 2955 . . . . . . . 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 14004 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
62613adantl3 1154 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  NN )
63 eluz2b3 11151 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
6463baib 901 . . . . . . . . 9  |-  ( ( A  gcd  B )  e.  NN  ->  (
( A  gcd  B
)  e.  ( ZZ>= ` 
2 )  <->  ( A  gcd  B )  =/=  1
) )
6562, 64syl 16 . . . . . . . 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 234 . . . . . . 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 14103 . . . . . . 7  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( A  gcd  B ) )
69 gcddvds 14005 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
7041, 21, 69syl2anc 661 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  A )
72 iddvdsexp 13861 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  A  ||  ( A ^ N ) )
7341, 42, 72syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  A  ||  ( A ^ N
) )
7462nnzd 10961 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
7574adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  e.  ZZ )
76 dvdstr 13871 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 679 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  ( A ^ N ) )
79 dvdstr 13871 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 674 . . . . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0
) )  /\  p  e.  Prime )  ->  ( A  gcd  B )  ||  B )
83 dvdstr 13871 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 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 674 . . . . . . . . . . 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 533 . . . . . . . . . 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 14033 . . . . . . . . . . 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 1228 . . . . . . . . . 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 4451 . . . . . . . . . . . . . 14  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  (
p  ||  ( ( A ^ N )  gcd 
B )  <->  p  ||  1
) )
9089notbid 294 . . . . . . . . . . . . 13  |-  ( ( ( A ^ N
)  gcd  B )  =  1  ->  ( -.  p  ||  ( ( A ^ N )  gcd  B )  <->  -.  p  ||  1 ) )
9153, 90syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  ( ( ( A ^ N
)  gcd  B )  =  1  ->  -.  p  ||  ( ( A ^ N )  gcd 
B ) ) )
9291necon2ad 2680 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  ( p 
||  ( ( A ^ N )  gcd 
B )  ->  (
( A ^ N
)  gcd  B )  =/=  1 ) )
9392adantl 466 . . . . . . . . . 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 55 . . . . . . . . 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 2955 . . . . . . . 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 11151 . . . . . . . . . 10  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( ( A ^ N )  gcd  B
)  e.  NN  /\  ( ( A ^ N )  gcd  B
)  =/=  1 ) )
9796baib 901 . . . . . . . . 9  |-  ( ( ( A ^ N
)  gcd  B )  e.  NN  ->  ( (
( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( A ^ N
)  gcd  B )  =/=  1 ) )
9834, 97syl 16 . . . . . . . 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 234 . . . . . . 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 191 . . . . 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 283 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =/=  1  <->  ( A  gcd  B )  =/=  1 ) )
103102necon4bid 2726 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( ( A ^ N )  gcd 
B )  =  1  <-> 
( A  gcd  B
)  =  1 ) )
104103ex 434 . 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 158 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( ( A ^ N )  gcd  B
)  =  1  <->  ( A  gcd  B )  =  1 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ^cexp 12129    || cdivides 13840    gcd cgcd 13996   Primecprime 14069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-dvds 13841  df-gcd 13997  df-prm 14070
This theorem is referenced by:  rpexp1i  14114  phiprmpw  14158  pockthlem  14275
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