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Theorem rpexp 13905
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 11997 . . . . . 6  |-  ( N  e.  NN  ->  (
0 ^ N )  =  0 )
21oveq1d 6202 . . . . 5  |-  ( N  e.  NN  ->  (
( 0 ^ N
)  gcd  0 )  =  ( 0  gcd  0 ) )
32eqeq1d 2453 . . . 4  |-  ( N  e.  NN  ->  (
( ( 0 ^ N )  gcd  0
)  =  1  <->  (
0  gcd  0 )  =  1 ) )
4 oveq1 6194 . . . . . . 7  |-  ( A  =  0  ->  ( A ^ N )  =  ( 0 ^ N
) )
5 oveq12 6196 . . . . . . 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 2453 . . . . 5  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( ( ( A ^ N )  gcd  B )  =  1  <->  ( ( 0 ^ N )  gcd  0 )  =  1 ) )
8 oveq12 6196 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  gcd  B )  =  ( 0  gcd  0 ) )
98eqeq1d 2453 . . . . 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 1011 . 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 13895 . . . . . . 7  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( A ^ N )  gcd 
B ) )
14 simpl1 991 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  ZZ )
15 simpl3 993 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN )
1615nnnn0d 10734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  N  e.  NN0 )
17 zexpcl 11978 . . . . . . . . . . . . . . . . 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 992 . . . . . . . . . . . . . . . 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 13798 . . . . . . . . . . . . . . 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 13866 . . . . . . . . . . . . . . 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 10846 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  A  e.  CC )
29 expeq0 11992 . . . . . . . . . . . . . . . . . . . 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 13797 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A ^ N )  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( ( A ^ N )  =  0  /\  B  =  0 ) )  ->  ( ( A ^ N )  gcd 
B )  e.  NN )
3418, 20, 32, 33syl21anc 1218 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( ( A ^ N )  gcd  B
)  e.  NN )
3534nnzd 10844 . . . . . . . . . . . . . . 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 13665 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . 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 13899 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  (
p  ||  ( A ^ N )  <->  p  ||  A
) )
4440, 41, 42, 43syl3anc 1219 . . . . . . . . . . . 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 13665 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 13826 . . . . . . . . . . 11  |-  ( ( p  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( p  ||  A  /\  p  ||  B )  ->  p  ||  ( A  gcd  B ) ) )
5226, 41, 21, 51syl3anc 1219 . . . . . . . . . 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 13896 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
54 breq2 4391 . . . . . . . . . . . . . 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 2659 . . . . . . . . . . 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 2934 . . . . . . . 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 13797 . . . . . . . . . 10  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
62613adantl3 1146 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN )  /\  -.  ( A  =  0  /\  B  =  0 ) )  -> 
( A  gcd  B
)  e.  NN )
63 eluz2b3 11026 . . . . . . . . . 10  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  <->  ( ( A  gcd  B )  e.  NN  /\  ( A  gcd  B )  =/=  1 ) )
6463baib 896 . . . . . . . . 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 13895 . . . . . . 7  |-  ( ( A  gcd  B )  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  ( A  gcd  B ) )
69 gcddvds 13798 . . . . . . . . . . . . . . 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 13655 . . . . . . . . . . . . . 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 10844 . . . . . . . . . . . . . . 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 13665 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 13665 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 13665 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 13826 . . . . . . . . . . 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 1219 . . . . . . . . . 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 4391 . . . . . . . . . . . . . 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 2659 . . . . . . . . . . 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 2934 . . . . . . . 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 11026 . . . . . . . . . 10  |-  ( ( ( A ^ N
)  gcd  B )  e.  ( ZZ>= `  2 )  <->  ( ( ( A ^ N )  gcd  B
)  e.  NN  /\  ( ( A ^ N )  gcd  B
)  =/=  1 ) )
9796baib 896 . . . . . . . . 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 2705 . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2642   E.wrex 2794   class class class wbr 4387   ` cfv 5513  (class class class)co 6187   CCcc 9378   0cc0 9380   1c1 9381   NNcn 10420   2c2 10469   NN0cn0 10677   ZZcz 10744   ZZ>=cuz 10959   ^cexp 11963    || cdivides 13634    gcd cgcd 13789   Primecprime 13862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-pre-sup 9458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-2o 7018  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-sup 7789  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-n0 10678  df-z 10745  df-uz 10960  df-rp 11090  df-fz 11536  df-fl 11740  df-mod 11807  df-seq 11905  df-exp 11964  df-cj 12687  df-re 12688  df-im 12689  df-sqr 12823  df-abs 12824  df-dvds 13635  df-gcd 13790  df-prm 13863
This theorem is referenced by:  rpexp1i  13906  phiprmpw  13950  pockthlem  14065
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