Theorem rpexp 12291

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.

Step	Hyp	Ref	Expression
1		0exp 10645	. . . . . 6 |- ( N e. NN -> ( 0 ^ N ) = 0 )
2	1	oveq1d 5933	. . . . 5 |- ( N e. NN -> ( ( 0 ^ N ) gcd 0 ) = ( 0 gcd 0 ) )
3	2	eqeq1d 2202	. . . 4 |- ( N e. NN -> ( ( ( 0 ^ N ) gcd 0 ) = 1 <-> ( 0 gcd 0 ) = 1 ) )
4		oveq1 5925	. . . . . . 7 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
5		oveq12 5927	. . . . . . 7 |- ( ( ( A ^ N ) = ( 0 ^ N ) /\ B = 0 ) -> ( ( A ^ N ) gcd B ) = ( ( 0 ^ N ) gcd 0 ) )
6	4, 5	sylan 283	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) -> ( ( A ^ N ) gcd B ) = ( ( 0 ^ N ) gcd 0 ) )
7	6	eqeq1d 2202	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( ( 0 ^ N ) gcd 0 ) = 1 ) )
8		oveq12 5927	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
9	8	eqeq1d 2202	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( A gcd B ) = 1 <-> ( 0 gcd 0 ) = 1 ) )
10	7, 9	bibi12d 235	. . . 4 |- ( ( A = 0 /\ B = 0 ) -> ( ( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) <-> ( ( ( 0 ^ N ) gcd 0 ) = 1 <-> ( 0 gcd 0 ) = 1 ) ) )
11	3, 10	syl5ibrcom 157	. . 3 |- ( N e. NN -> ( ( A = 0 /\ B = 0 ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) ) )
12	11	3ad2ant3 1022	. 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 12276	. . . . . . 7 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( ( A ^ N ) gcd B ) )
14		simpl1 1002	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. ZZ )
15		simpl3 1004	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN )
16	15	nnnn0d 9293	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN0 )
17		zexpcl 10625	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )
18	14, 16, 17	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A ^ N ) e. ZZ )
19	18	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( A ^ N ) e. ZZ )
20		simpl2 1003	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> B e. ZZ )
21	20	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> B e. ZZ )
22		gcddvds 12100	. . . . . . . . . . . . . . 15 |- ( ( ( A ^ N ) e. ZZ /\ B e. ZZ ) -> ( ( ( A ^ N ) gcd B ) || ( A ^ N ) /\ ( ( A ^ N ) gcd B ) || B ) )
23	19, 21, 22	syl2anc 411	. . . . . . . . . . . . . 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 ) )
24	23	simpld 112	. . . . . . . . . . . . 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 12249	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
26	25	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> p e. ZZ )
27		simpr 110	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> -. ( A = 0 /\ B = 0 ) )
28	14	zcnd 9440	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. CC )
29		expeq0 10641	. . . . . . . . . . . . . . . . . . . 20 |- ( ( A e. CC /\ N e. NN ) -> ( ( A ^ N ) = 0 <-> A = 0 ) )
30	28, 15, 29	syl2anc 411	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( A ^ N ) = 0 <-> A = 0 ) )
31	30	anbi1d 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( ( A ^ N ) = 0 /\ B = 0 ) <-> ( A = 0 /\ B = 0 ) ) )
32	27, 31	mtbird 674	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> -. ( ( A ^ N ) = 0 /\ B = 0 ) )
33		gcdn0cl 12099	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A ^ N ) e. ZZ /\ B e. ZZ ) /\ -. ( ( A ^ N ) = 0 /\ B = 0 ) ) -> ( ( A ^ N ) gcd B ) e. NN )
34	18, 20, 32, 33	syl21anc 1248	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( A ^ N ) gcd B ) e. NN )
35	34	nnzd 9438	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( A ^ N ) gcd B ) e. ZZ )
36	35	adantr 276	. . . . . . . . . . . . . 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 11971	. . . . . . . . . . . . . 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 ) ) )
38	26, 36, 19, 37	syl3anc 1249	. . . . . . . . . . . . 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 ) ) )
39	24, 38	mpan2d 428	. . . . . . . . . . . 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 110	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> p e. Prime )
41		simpll1 1038	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> A e. ZZ )
42	15	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> N e. NN )
43		prmdvdsexp 12286	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ A e. ZZ /\ N e. NN ) -> ( p || ( A ^ N ) <-> p || A ) )
44	40, 41, 42, 43	syl3anc 1249	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( p || ( A ^ N ) <-> p || A ) )
45	39, 44	sylibd 149	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( p || ( ( A ^ N ) gcd B ) -> p || A ) )
46	23	simprd 114	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( ( A ^ N ) gcd B ) || B )
47		dvdstr 11971	. . . . . . . . . . . . 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 ) )
48	26, 36, 21, 47	syl3anc 1249	. . . . . . . . . . . 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 ) )
49	46, 48	mpan2d 428	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( p || ( ( A ^ N ) gcd B ) -> p || B ) )
50	45, 49	jcad 307	. . . . . . . . . 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 12149	. . . . . . . . . . 11 |- ( ( p e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( p || A /\ p || B ) -> p || ( A gcd B ) ) )
52	26, 41, 21, 51	syl3anc 1249	. . . . . . . . . 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 12278	. . . . . . . . . . . . 13 |- ( p e. Prime -> -. p || 1 )
54		breq2 4033	. . . . . . . . . . . . . 14 |- ( ( A gcd B ) = 1 -> ( p || ( A gcd B ) <-> p || 1 ) )
55	54	notbid 668	. . . . . . . . . . . . 13 |- ( ( A gcd B ) = 1 -> ( -. p || ( A gcd B ) <-> -. p || 1 ) )
56	53, 55	syl5ibrcom 157	. . . . . . . . . . . 12 |- ( p e. Prime -> ( ( A gcd B ) = 1 -> -. p || ( A gcd B ) ) )
57	56	necon2ad 2421	. . . . . . . . . . 11 |- ( p e. Prime -> ( p || ( A gcd B ) -> ( A gcd B ) =/= 1 ) )
58	57	adantl 277	. . . . . . . . . 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 ) )
59	50, 52, 58	3syld 57	. . . . . . . . 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 ) )
60	59	rexlimdva 2611	. . . . . . . 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 12099	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
62	61	3adantl3 1157	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
63		eluz2b3 9669	. . . . . . . . . 10 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
64	63	baib 920	. . . . . . . . 9 |- ( ( A gcd B ) e. NN -> ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( A gcd B ) =/= 1 ) )
65	62, 64	syl 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 ) )
66	60, 65	sylibrd 169	. . . . . . 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 ) ) )
67	13, 66	syl5 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 12276	. . . . . . 7 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( A gcd B ) )
69		gcddvds 12100	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
70	41, 21, 69	syl2anc 411	. . . . . . . . . . . . . 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 ) )
71	70	simpld 112	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( A gcd B ) || A )
72		iddvdsexp 11958	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ N e. NN ) -> A || ( A ^ N ) )
73	41, 42, 72	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> A || ( A ^ N ) )
74	62	nnzd 9438	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. ZZ )
75	74	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( A gcd B ) e. ZZ )
76		dvdstr 11971	. . . . . . . . . . . . . 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 ) ) )
77	75, 41, 19, 76	syl3anc 1249	. . . . . . . . . . . . 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 ) ) )
78	71, 73, 77	mp2and 433	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( A gcd B ) || ( A ^ N ) )
79		dvdstr 11971	. . . . . . . . . . . . 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 ) ) )
80	26, 75, 19, 79	syl3anc 1249	. . . . . . . . . . . 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 ) ) )
81	78, 80	mpan2d 428	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( p || ( A gcd B ) -> p || ( A ^ N ) ) )
82	70	simprd 114	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( A gcd B ) || B )
83		dvdstr 11971	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ ( A gcd B ) e. ZZ /\ B e. ZZ ) -> ( ( p || ( A gcd B ) /\ ( A gcd B ) || B ) -> p || B ) )
84	26, 75, 21, 83	syl3anc 1249	. . . . . . . . . . . 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 ) )
85	82, 84	mpan2d 428	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) /\ p e. Prime ) -> ( p || ( A gcd B ) -> p || B ) )
86	81, 85	jcad 307	. . . . . . . . . 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 12149	. . . . . . . . . . 11 |- ( ( p e. ZZ /\ ( A ^ N ) e. ZZ /\ B e. ZZ ) -> ( ( p || ( A ^ N ) /\ p || B ) -> p || ( ( A ^ N ) gcd B ) ) )
88	26, 19, 21, 87	syl3anc 1249	. . . . . . . . . 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 4033	. . . . . . . . . . . . . 14 |- ( ( ( A ^ N ) gcd B ) = 1 -> ( p || ( ( A ^ N ) gcd B ) <-> p || 1 ) )
90	89	notbid 668	. . . . . . . . . . . . 13 |- ( ( ( A ^ N ) gcd B ) = 1 -> ( -. p || ( ( A ^ N ) gcd B ) <-> -. p || 1 ) )
91	53, 90	syl5ibrcom 157	. . . . . . . . . . . 12 |- ( p e. Prime -> ( ( ( A ^ N ) gcd B ) = 1 -> -. p || ( ( A ^ N ) gcd B ) ) )
92	91	necon2ad 2421	. . . . . . . . . . 11 |- ( p e. Prime -> ( p || ( ( A ^ N ) gcd B ) -> ( ( A ^ N ) gcd B ) =/= 1 ) )
93	92	adantl 277	. . . . . . . . . 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 ) )
94	86, 88, 93	3syld 57	. . . . . . . . 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 ) )
95	94	rexlimdva 2611	. . . . . . . 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 9669	. . . . . . . . . 10 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( ( A ^ N ) gcd B ) e. NN /\ ( ( A ^ N ) gcd B ) =/= 1 ) )
97	96	baib 920	. . . . . . . . 9 |- ( ( ( A ^ N ) gcd B ) e. NN -> ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A ^ N ) gcd B ) =/= 1 ) )
98	34, 97	syl 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 ) )
99	95, 98	sylibrd 169	. . . . . . 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 ) ) )
100	68, 99	syl5 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 ) ) )
101	67, 100	impbid 129	. . . . 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 ) ) )
102	101, 98, 65	3bitr3d 218	. . . 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 999	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> A e. ZZ )
104		simp3 1001	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> N e. NN )
105	104	nnnn0d 9293	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> N e. NN0 )
106	103, 105, 17	syl2anc 411	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A ^ N ) e. ZZ )
107		simp2 1000	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> B e. ZZ )
108	106, 107	gcdcld 12105	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. NN0 )
109	108	nn0zd 9437	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. ZZ )
110		1zzd 9344	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> 1 e. ZZ )
111		zdceq 9392	. . . . . . 7 |- ( ( ( ( A ^ N ) gcd B ) e. ZZ /\ 1 e. ZZ ) -> DECID ( ( A ^ N ) gcd B ) = 1 )
112	109, 110, 111	syl2anc 411	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> DECID ( ( A ^ N ) gcd B ) = 1 )
113	103, 107	gcdcld 12105	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. NN0 )
114	113	nn0zd 9437	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. ZZ )
115		zdceq 9392	. . . . . . 7 |- ( ( ( A gcd B ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A gcd B ) = 1 )
116	114, 110, 115	syl2anc 411	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> DECID ( A gcd B ) = 1 )
117		nebidc 2444	. . . . . 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 ) ) ) )
118	112, 116, 117	sylc 62	. . . . 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 ) ) )
119	118	adantr 276	. . . 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 ) ) )
120	102, 119	mpbird 167	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) )
121	120	ex 115	. 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 12081	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID ( A = 0 /\ B = 0 ) )
123		exmiddc 837	. . . 4 |- (DECID ( A = 0 /\ B = 0 ) -> ( ( A = 0 /\ B = 0 ) \/ -. ( A = 0 /\ B = 0 ) ) )
124	122, 123	syl 14	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A = 0 /\ B = 0 ) \/ -. ( A = 0 /\ B = 0 ) ) )
125	124	3adant3 1019	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A = 0 /\ B = 0 ) \/ -. ( A = 0 /\ B = 0 ) ) )
126	12, 121, 125	mpjaod 719	1 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( A gcd B ) = 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   E.wrex 2473   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   ^cexp 10609   || cdvds 11930   gcd cgcd 12079   Primecprime 12245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246

This theorem is referenced by:  rpexp1i  12292  phiprmpw  12360  pockthlem  12494  logbgcd1irr  15099  logbgcd1irraplemexp  15100