Theorem rpexp 12550

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 10741	. . . . . 6 |- ( N e. NN -> ( 0 ^ N ) = 0 )
2	1	oveq1d 5972	. . . . 5 |- ( N e. NN -> ( ( 0 ^ N ) gcd 0 ) = ( 0 gcd 0 ) )
3	2	eqeq1d 2215	. . . 4 |- ( N e. NN -> ( ( ( 0 ^ N ) gcd 0 ) = 1 <-> ( 0 gcd 0 ) = 1 ) )
4		oveq1 5964	. . . . . . 7 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
5		oveq12 5966	. . . . . . 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 2215	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( ( 0 ^ N ) gcd 0 ) = 1 ) )
8		oveq12 5966	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
9	8	eqeq1d 2215	. . . . 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 1023	. 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 12535	. . . . . . 7 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( ( A ^ N ) gcd B ) )
14		simpl1 1003	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. ZZ )
15		simpl3 1005	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN )
16	15	nnnn0d 9368	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN0 )
17		zexpcl 10721	. . . . . . . . . . . . . . . . 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 1004	. . . . . . . . . . . . . . . 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 12359	. . . . . . . . . . . . . . 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 12508	. . . . . . . . . . . . . . 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 9516	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. CC )
29		expeq0 10737	. . . . . . . . . . . . . . . . . . . 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 675	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> -. ( ( A ^ N ) = 0 /\ B = 0 ) )
33		gcdn0cl 12358	. . . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( A ^ N ) gcd B ) e. NN )
35	34	nnzd 9514	. . . . . . . . . . . . . . 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 12214	. . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . 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 1039	. . . . . . . . . . . . 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 12545	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ A e. ZZ /\ N e. NN ) -> ( p || ( A ^ N ) <-> p || A ) )
44	40, 41, 42, 43	syl3anc 1250	. . . . . . . . . . . 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 12214	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 12408	. . . . . . . . . . 11 |- ( ( p e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( p || A /\ p || B ) -> p || ( A gcd B ) ) )
52	26, 41, 21, 51	syl3anc 1250	. . . . . . . . . 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 12537	. . . . . . . . . . . . 13 |- ( p e. Prime -> -. p || 1 )
54		breq2 4055	. . . . . . . . . . . . . 14 |- ( ( A gcd B ) = 1 -> ( p || ( A gcd B ) <-> p || 1 ) )
55	54	notbid 669	. . . . . . . . . . . . 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 2434	. . . . . . . . . . 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 2624	. . . . . . . 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 12358	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
62	61	3adantl3 1158	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
63		eluz2b3 9745	. . . . . . . . . 10 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
64	63	baib 921	. . . . . . . . 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 12535	. . . . . . 7 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( A gcd B ) )
69		gcddvds 12359	. . . . . . . . . . . . . . 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 12201	. . . . . . . . . . . . . 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 9514	. . . . . . . . . . . . . . 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 12214	. . . . . . . . . . . . . 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 1250	. . . . . . . . . . . . 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 12214	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 12214	. . . . . . . . . . . . 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 1250	. . . . . . . . . . . 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 12408	. . . . . . . . . . 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 1250	. . . . . . . . . 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 4055	. . . . . . . . . . . . . 14 |- ( ( ( A ^ N ) gcd B ) = 1 -> ( p || ( ( A ^ N ) gcd B ) <-> p || 1 ) )
90	89	notbid 669	. . . . . . . . . . . . 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 2434	. . . . . . . . . . 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 2624	. . . . . . . 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 9745	. . . . . . . . . 10 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( ( A ^ N ) gcd B ) e. NN /\ ( ( A ^ N ) gcd B ) =/= 1 ) )
97	96	baib 921	. . . . . . . . 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 1000	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> A e. ZZ )
104		simp3 1002	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> N e. NN )
105	104	nnnn0d 9368	. . . . . . . . . 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 1001	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> B e. ZZ )
108	106, 107	gcdcld 12364	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. NN0 )
109	108	nn0zd 9513	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. ZZ )
110		1zzd 9419	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> 1 e. ZZ )
111		zdceq 9468	. . . . . . 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 12364	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. NN0 )
114	113	nn0zd 9513	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. ZZ )
115		zdceq 9468	. . . . . . 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 2457	. . . . . 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 12351	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID ( A = 0 /\ B = 0 ) )
123		exmiddc 838	. . . 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 1020	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A = 0 /\ B = 0 ) \/ -. ( A = 0 /\ B = 0 ) ) )
126	12, 121, 125	mpjaod 720	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   E.wrex 2486   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   CCcc 7943   0cc0 7945   1c1 7946   NNcn 9056   2c2 9107   NN0cn0 9315   ZZcz 9392   ZZ>=cuz 9668   ^cexp 10705   || cdvds 12173   gcd cgcd 12349   Primecprime 12504

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-1o 6515  df-2o 6516  df-er 6633  df-en 6841  df-sup 7101  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-fl 10435  df-mod 10490  df-seqfrec 10615  df-exp 10706  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-dvds 12174  df-gcd 12350  df-prm 12505

This theorem is referenced by:  rpexp1i  12551  phiprmpw  12619  pockthlem  12754  logbgcd1irr  15514  logbgcd1irraplemexp  15515