Theorem rpexp 12417

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 10717	. . . . . 6 |- ( N e. NN -> ( 0 ^ N ) = 0 )
2	1	oveq1d 5958	. . . . 5 |- ( N e. NN -> ( ( 0 ^ N ) gcd 0 ) = ( 0 gcd 0 ) )
3	2	eqeq1d 2213	. . . 4 |- ( N e. NN -> ( ( ( 0 ^ N ) gcd 0 ) = 1 <-> ( 0 gcd 0 ) = 1 ) )
4		oveq1 5950	. . . . . . 7 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
5		oveq12 5952	. . . . . . 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 2213	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( ( 0 ^ N ) gcd 0 ) = 1 ) )
8		oveq12 5952	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
9	8	eqeq1d 2213	. . . . 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 12402	. . . . . . 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 9347	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN0 )
17		zexpcl 10697	. . . . . . . . . . . . . . . . 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 12226	. . . . . . . . . . . . . . 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 12375	. . . . . . . . . . . . . . 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 9495	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. CC )
29		expeq0 10713	. . . . . . . . . . . . . . . . . . . 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 12225	. . . . . . . . . . . . . . . . 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 9493	. . . . . . . . . . . . . . 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 12081	. . . . . . . . . . . . . 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 12412	. . . . . . . . . . . . 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 12081	. . . . . . . . . . . . 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 12275	. . . . . . . . . . 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 12404	. . . . . . . . . . . . 13 |- ( p e. Prime -> -. p || 1 )
54		breq2 4047	. . . . . . . . . . . . . 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 2432	. . . . . . . . . . 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 2622	. . . . . . . 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 12225	. . . . . . . . . 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 9724	. . . . . . . . . 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 12402	. . . . . . 7 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( A gcd B ) )
69		gcddvds 12226	. . . . . . . . . . . . . . 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 12068	. . . . . . . . . . . . . 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 9493	. . . . . . . . . . . . . . 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 12081	. . . . . . . . . . . . . 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 12081	. . . . . . . . . . . . 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 12081	. . . . . . . . . . . . 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 12275	. . . . . . . . . . 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 4047	. . . . . . . . . . . . . 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 2432	. . . . . . . . . . 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 2622	. . . . . . . 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 9724	. . . . . . . . . 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 9347	. . . . . . . . . 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 12231	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. NN0 )
109	108	nn0zd 9492	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. ZZ )
110		1zzd 9398	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> 1 e. ZZ )
111		zdceq 9447	. . . . . . 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 12231	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. NN0 )
114	113	nn0zd 9492	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. ZZ )
115		zdceq 9447	. . . . . . 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 2455	. . . . . 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 12218	. . . 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 1372   e. wcel 2175   =/= wne 2375   E.wrex 2484   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   CCcc 7922   0cc0 7924   1c1 7925   NNcn 9035   2c2 9086   NN0cn0 9294   ZZcz 9371   ZZ>=cuz 9647   ^cexp 10681   || cdvds 12040   gcd cgcd 12216   Primecprime 12371

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-1o 6501  df-2o 6502  df-er 6619  df-en 6827  df-sup 7085  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-dvds 12041  df-gcd 12217  df-prm 12372

This theorem is referenced by:  rpexp1i  12418  phiprmpw  12486  pockthlem  12621  logbgcd1irr  15381  logbgcd1irraplemexp  15382