Theorem rpexp 12724

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 10835	. . . . . 6 |- ( N e. NN -> ( 0 ^ N ) = 0 )
2	1	oveq1d 6032	. . . . 5 |- ( N e. NN -> ( ( 0 ^ N ) gcd 0 ) = ( 0 gcd 0 ) )
3	2	eqeq1d 2240	. . . 4 |- ( N e. NN -> ( ( ( 0 ^ N ) gcd 0 ) = 1 <-> ( 0 gcd 0 ) = 1 ) )
4		oveq1 6024	. . . . . . 7 |- ( A = 0 -> ( A ^ N ) = ( 0 ^ N ) )
5		oveq12 6026	. . . . . . 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 2240	. . . . 5 |- ( ( A = 0 /\ B = 0 ) -> ( ( ( A ^ N ) gcd B ) = 1 <-> ( ( 0 ^ N ) gcd 0 ) = 1 ) )
8		oveq12 6026	. . . . . 6 |- ( ( A = 0 /\ B = 0 ) -> ( A gcd B ) = ( 0 gcd 0 ) )
9	8	eqeq1d 2240	. . . . 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 1046	. 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 12709	. . . . . . 7 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( ( A ^ N ) gcd B ) )
14		simpl1 1026	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. ZZ )
15		simpl3 1028	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN )
16	15	nnnn0d 9454	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> N e. NN0 )
17		zexpcl 10815	. . . . . . . . . . . . . . . . 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 1027	. . . . . . . . . . . . . . . 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 12533	. . . . . . . . . . . . . . 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 12682	. . . . . . . . . . . . . . 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 9602	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> A e. CC )
29		expeq0 10831	. . . . . . . . . . . . . . . . . . . 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 679	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> -. ( ( A ^ N ) = 0 /\ B = 0 ) )
33		gcdn0cl 12532	. . . . . . . . . . . . . . . . 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 1272	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( ( A ^ N ) gcd B ) e. NN )
35	34	nnzd 9600	. . . . . . . . . . . . . . 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 12388	. . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . 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 1062	. . . . . . . . . . . . 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 12719	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ A e. ZZ /\ N e. NN ) -> ( p || ( A ^ N ) <-> p || A ) )
44	40, 41, 42, 43	syl3anc 1273	. . . . . . . . . . . 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 12388	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 12582	. . . . . . . . . . 11 |- ( ( p e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( p || A /\ p || B ) -> p || ( A gcd B ) ) )
52	26, 41, 21, 51	syl3anc 1273	. . . . . . . . . 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 12711	. . . . . . . . . . . . 13 |- ( p e. Prime -> -. p || 1 )
54		breq2 4092	. . . . . . . . . . . . . 14 |- ( ( A gcd B ) = 1 -> ( p || ( A gcd B ) <-> p || 1 ) )
55	54	notbid 673	. . . . . . . . . . . . 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 2459	. . . . . . . . . . 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 2650	. . . . . . . 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 12532	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
62	61	3adantl3 1181	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
63		eluz2b3 9837	. . . . . . . . . 10 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( A gcd B ) e. NN /\ ( A gcd B ) =/= 1 ) )
64	63	baib 926	. . . . . . . . 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 12709	. . . . . . 7 |- ( ( A gcd B ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( A gcd B ) )
69		gcddvds 12533	. . . . . . . . . . . . . . 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 12375	. . . . . . . . . . . . . 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 9600	. . . . . . . . . . . . . . 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 12388	. . . . . . . . . . . . . 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 1273	. . . . . . . . . . . . 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 12388	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 12388	. . . . . . . . . . . . 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 1273	. . . . . . . . . . . 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 12582	. . . . . . . . . . 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 1273	. . . . . . . . . 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 4092	. . . . . . . . . . . . . 14 |- ( ( ( A ^ N ) gcd B ) = 1 -> ( p || ( ( A ^ N ) gcd B ) <-> p || 1 ) )
90	89	notbid 673	. . . . . . . . . . . . 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 2459	. . . . . . . . . . 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 2650	. . . . . . . 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 9837	. . . . . . . . . 10 |- ( ( ( A ^ N ) gcd B ) e. ( ZZ>= ` 2 ) <-> ( ( ( A ^ N ) gcd B ) e. NN /\ ( ( A ^ N ) gcd B ) =/= 1 ) )
97	96	baib 926	. . . . . . . . 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 1023	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> A e. ZZ )
104		simp3 1025	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> N e. NN )
105	104	nnnn0d 9454	. . . . . . . . . 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 1024	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> B e. ZZ )
108	106, 107	gcdcld 12538	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. NN0 )
109	108	nn0zd 9599	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A ^ N ) gcd B ) e. ZZ )
110		1zzd 9505	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> 1 e. ZZ )
111		zdceq 9554	. . . . . . 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 12538	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. NN0 )
114	113	nn0zd 9599	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( A gcd B ) e. ZZ )
115		zdceq 9554	. . . . . . 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 2482	. . . . . 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 12525	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID ( A = 0 /\ B = 0 ) )
123		exmiddc 843	. . . 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 1043	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N e. NN ) -> ( ( A = 0 /\ B = 0 ) \/ -. ( A = 0 /\ B = 0 ) ) )
126	12, 121, 125	mpjaod 725	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   NNcn 9142   2c2 9193   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   ^cexp 10799   || cdvds 12347   gcd cgcd 12523   Primecprime 12678

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679

This theorem is referenced by:  rpexp1i  12725  phiprmpw  12793  pockthlem  12928  logbgcd1irr  15690  logbgcd1irraplemexp  15691