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Theorem rpexp 11014
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 9889 . . . . . 6  |-  ( N  e.  NN  ->  (
0 ^ N )  =  0 )
21oveq1d 5628 . . . . 5  |-  ( N  e.  NN  ->  (
( 0 ^ N
)  gcd  0 )  =  ( 0  gcd  0 ) )
32eqeq1d 2093 . . . 4  |-  ( N  e.  NN  ->  (
( ( 0 ^ N )  gcd  0
)  =  1  <->  (
0  gcd  0 )  =  1 ) )
4 oveq1 5620 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
5 oveq12 5622 . . . . . . 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 2093 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( ( 0 ^ N )  gcd  0 )  =  1 ) )
8 oveq12 5622 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
98eqeq1d 2093 . . . . 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 964 . 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 11001 . . . . . . 7  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( A ^ N )  gcd 
B ) )
14 simpl1 944 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
15 simpl3 946 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN )
1615nnnn0d 8659 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN0 )
17 zexpcl 9869 . . . . . . . . . . . . . . . . 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 945 . . . . . . . . . . . . . . . 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 10837 . . . . . . . . . . . . . . 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 10975 . . . . . . . . . . . . . . 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 8802 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  CC )
29 expeq0 9885 . . . . . . . . . . . . . . . . . . . 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 10836 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )  ->  ( ( A ^ N )  gcd 
B )  e.  NN )
3418, 20, 32, 33syl21anc 1171 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  NN )
3534nnzd 8800 . . . . . . . . . . . . . . 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 10715 . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . 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 980 . . . . . . . . . . . . 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 11009 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4440, 41, 42, 43syl3anc 1172 . . . . . . . . . . . 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 10715 . . . . . . . . . . . . 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 1172 . . . . . . . . . . . 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 10883 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
5226, 41, 21, 51syl3anc 1172 . . . . . . . . . 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 11003 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
54 breq2 3824 . . . . . . . . . . . . . 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 2308 . . . . . . . . . . 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 2485 . . . . . . . 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 10836 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
62613adantl3 1099 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  NN )
63 eluz2b3 9023 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
6463baib 864 . . . . . . . . 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 11001 . . . . . . 7  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( A  gcd  B ) )
69 gcddvds 10837 . . . . . . . . . . . . . . 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 10702 . . . . . . . . . . . . . 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 8800 . . . . . . . . . . . . . . 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 10715 . . . . . . . . . . . . . 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 1172 . . . . . . . . . . . . 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 10715 . . . . . . . . . . . . 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 1172 . . . . . . . . . . . 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 10715 . . . . . . . . . . . . 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 1172 . . . . . . . . . . . 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 10883 . . . . . . . . . . 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 1172 . . . . . . . . . 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 3824 . . . . . . . . . . . . . 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 2308 . . . . . . . . . . 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 2485 . . . . . . . 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 9023 . . . . . . . . . 10  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( ( A ^ N )  gcd  B
)  e.  NN  /\  ( ( A ^ N )  gcd  B
)  =/=  1 ) )
9796baib 864 . . . . . . . . 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 941 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  A  e.  ZZ )
104 simp3 943 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  N  e.  NN )
105104nnnn0d 8659 . . . . . . . . . 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 942 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  B  e.  ZZ )
108106, 107gcdcld 10842 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A ^ N
)  gcd  B )  e.  NN0 )
109108nn0zd 8799 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  (
( A ^ N
)  gcd  B )  e.  ZZ )
110 1zzd 8710 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  1  e.  ZZ )
111 zdceq 8755 . . . . . . 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 10842 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  gcd  B )  e. 
NN0 )
114113nn0zd 8799 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  ->  ( A  gcd  B )  e.  ZZ )
115 zdceq 8755 . . . . . . 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 2331 . . . . . 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 10822 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  ( A  =  0  /\  B  =  0 ) )
123 exmiddc 780 . . . 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 961 . 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 778    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   E.wrex 2356   class class class wbr 3820   ` cfv 4981  (class class class)co 5613   CCcc 7292   0cc0 7294   1c1 7295   NNcn 8357   2c2 8407   NN0cn0 8606   ZZcz 8683   ZZ>=cuz 8951   ^cexp 9853    || cdvds 10678    gcd cgcd 10820   Primecprime 10971
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407  ax-arch 7408  ax-caucvg 7409
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-ilim 4170  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-frec 6110  df-1o 6135  df-2o 6136  df-er 6244  df-en 6410  df-sup 6623  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-div 8079  df-inn 8358  df-2 8416  df-3 8417  df-4 8418  df-n0 8607  df-z 8684  df-uz 8952  df-q 9037  df-rp 9067  df-fz 9357  df-fzo 9482  df-fl 9605  df-mod 9658  df-iseq 9780  df-iexp 9854  df-cj 10172  df-re 10173  df-im 10174  df-rsqrt 10327  df-abs 10328  df-dvds 10679  df-gcd 10821  df-prm 10972
This theorem is referenced by:  rpexp1i  11015  phiprmpw  11080
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