Theorem divgcdcoprmex 12102

Description: Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.)

Assertion

Ref	Expression
divgcdcoprmex	|- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) )

Distinct variable groups:   A, a, b   B, a, b   M, a, b

Proof of Theorem divgcdcoprmex

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. ZZ )
2	1	anim2i 342	. . . 4 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A e. ZZ /\ B e. ZZ ) )
3		zeqzmulgcd 11971	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> E. a e. ZZ A = ( a x. ( A gcd B ) ) )
4	2, 3	syl 14	. . 3 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> E. a e. ZZ A = ( a x. ( A gcd B ) ) )
5	4	3adant3 1017	. 2 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. a e. ZZ A = ( a x. ( A gcd B ) ) )
6		zeqzmulgcd 11971	. . . . 5 |- ( ( B e. ZZ /\ A e. ZZ ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
7	6	adantlr 477	. . . 4 |- ( ( ( B e. ZZ /\ B =/= 0 ) /\ A e. ZZ ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
8	7	ancoms 268	. . 3 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
9	8	3adant3 1017	. 2 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
10		reeanv 2647	. . 3 |- ( E. a e. ZZ E. b e. ZZ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) <-> ( E. a e. ZZ A = ( a x. ( A gcd B ) ) /\ E. b e. ZZ B = ( b x. ( B gcd A ) ) ) )
11		zcn 9258	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
12	11	adantl 277	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> a e. CC )
13		gcdcl 11967	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
14	2, 13	syl 14	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) e. NN0 )
15	14	nn0cnd 9231	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) e. CC )
16	15	3adant3 1017	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. CC )
17	16	adantr 276	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> ( A gcd B ) e. CC )
18	12, 17	mulcomd 7979	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> ( a x. ( A gcd B ) ) = ( ( A gcd B ) x. a ) )
19		simp3 999	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> M = ( A gcd B ) )
20	19	eqcomd 2183	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) = M )
21	20	oveq1d 5890	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( ( A gcd B ) x. a ) = ( M x. a ) )
22	21	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> ( ( A gcd B ) x. a ) = ( M x. a ) )
23	18, 22	eqtrd 2210	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> ( a x. ( A gcd B ) ) = ( M x. a ) )
24	23	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> ( a x. ( A gcd B ) ) = ( M x. a ) )
25		eqeq1 2184	. . . . . . . . . 10 |- ( A = ( a x. ( A gcd B ) ) -> ( A = ( M x. a ) <-> ( a x. ( A gcd B ) ) = ( M x. a ) ) )
26	25	adantr 276	. . . . . . . . 9 |- ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> ( A = ( M x. a ) <-> ( a x. ( A gcd B ) ) = ( M x. a ) ) )
27	26	adantl 277	. . . . . . . 8 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> ( A = ( M x. a ) <-> ( a x. ( A gcd B ) ) = ( M x. a ) ) )
28	24, 27	mpbird 167	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> A = ( M x. a ) )
29		simpr 110	. . . . . . . 8 |- ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> B = ( b x. ( B gcd A ) ) )
30	2	ancomd 267	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( B e. ZZ /\ A e. ZZ ) )
31		gcdcom 11974	. . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ A e. ZZ ) -> ( B gcd A ) = ( A gcd B ) )
32	30, 31	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( B gcd A ) = ( A gcd B ) )
33	32	3adant3 1017	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( B gcd A ) = ( A gcd B ) )
34	33	oveq2d 5891	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( b x. ( B gcd A ) ) = ( b x. ( A gcd B ) ) )
35	34	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( b x. ( B gcd A ) ) = ( b x. ( A gcd B ) ) )
36		zcn 9258	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
37	36	adantl 277	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> b e. CC )
38	14	3adant3 1017	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
39	38	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( A gcd B ) e. NN0 )
40	39	nn0cnd 9231	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( A gcd B ) e. CC )
41	37, 40	mulcomd 7979	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( b x. ( A gcd B ) ) = ( ( A gcd B ) x. b ) )
42	20	adantr 276	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( A gcd B ) = M )
43	42	oveq1d 5890	. . . . . . . . . 10 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( ( A gcd B ) x. b ) = ( M x. b ) )
44	35, 41, 43	3eqtrd 2214	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( b x. ( B gcd A ) ) = ( M x. b ) )
45	44	adantlr 477	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( b x. ( B gcd A ) ) = ( M x. b ) )
46	29, 45	sylan9eqr 2232	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> B = ( M x. b ) )
47		zcn 9258	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> A e. CC )
48	47	3ad2ant1 1018	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. CC )
49	48	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> A e. CC )
50	12	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> a e. CC )
51		simp1 997	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. ZZ )
52	1	3ad2ant2 1019	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. ZZ )
53	51, 52	gcdcld 11969	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
54	53	nn0cnd 9231	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. CC )
55	54	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) e. CC )
56		gcdeq0 11978	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
57		simpr 110	. . . . . . . . . . . . . . . . . 18 |- ( ( A = 0 /\ B = 0 ) -> B = 0 )
58	56, 57	biimtrdi 163	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 -> B = 0 ) )
59	58	necon3d 2391	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B =/= 0 -> ( A gcd B ) =/= 0 ) )
60	59	impr 379	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) =/= 0 )
61	60	3adant3 1017	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) =/= 0 )
62	61	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) =/= 0 )
63	38	ad2antrr 488	. . . . . . . . . . . . . . 15 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) e. NN0 )
64	63	nn0zd 9373	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) e. ZZ )
65		0zd 9265	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> 0 e. ZZ )
66		zapne 9327	. . . . . . . . . . . . . 14 |- ( ( ( A gcd B ) e. ZZ /\ 0 e. ZZ ) -> ( ( A gcd B ) # 0 <-> ( A gcd B ) =/= 0 ) )
67	64, 65, 66	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( A gcd B ) # 0 <-> ( A gcd B ) =/= 0 ) )
68	62, 67	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) # 0 )
69	49, 50, 55, 68	divmulap3d 8782	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( A / ( A gcd B ) ) = a <-> A = ( a x. ( A gcd B ) ) ) )
70	69	bicomd 141	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A = ( a x. ( A gcd B ) ) <-> ( A / ( A gcd B ) ) = a ) )
71		zcn 9258	. . . . . . . . . . . . . . 15 |- ( B e. ZZ -> B e. CC )
72	71	adantr 276	. . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. CC )
73	72	3ad2ant2 1019	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. CC )
74	73	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> B e. CC )
75	36	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> b e. CC )
76	74, 75, 55, 68	divmulap3d 8782	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( B / ( A gcd B ) ) = b <-> B = ( b x. ( A gcd B ) ) ) )
77	2	3adant3 1017	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ ) )
78		gcdcom 11974	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
79	77, 78	syl 14	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) = ( B gcd A ) )
80	79	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
81	80	oveq2d 5891	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( b x. ( A gcd B ) ) = ( b x. ( B gcd A ) ) )
82	81	eqeq2d 2189	. . . . . . . . . . 11 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( B = ( b x. ( A gcd B ) ) <-> B = ( b x. ( B gcd A ) ) ) )
83	76, 82	bitr2d 189	. . . . . . . . . 10 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( B = ( b x. ( B gcd A ) ) <-> ( B / ( A gcd B ) ) = b ) )
84	70, 83	anbi12d 473	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) <-> ( ( A / ( A gcd B ) ) = a /\ ( B / ( A gcd B ) ) = b ) ) )
85		3anass 982	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) <-> ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) )
86	85	biimpri 133	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
87	86	3adant3 1017	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
88		divgcdcoprm0 12101	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
89	87, 88	syl 14	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
90		oveq12 5884	. . . . . . . . . . . 12 |- ( ( ( A / ( A gcd B ) ) = a /\ ( B / ( A gcd B ) ) = b ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = ( a gcd b ) )
91	90	eqeq1d 2186	. . . . . . . . . . 11 |- ( ( ( A / ( A gcd B ) ) = a /\ ( B / ( A gcd B ) ) = b ) -> ( ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 <-> ( a gcd b ) = 1 ) )
92	89, 91	syl5ibcom 155	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( ( ( A / ( A gcd B ) ) = a /\ ( B / ( A gcd B ) ) = b ) -> ( a gcd b ) = 1 ) )
93	92	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( ( A / ( A gcd B ) ) = a /\ ( B / ( A gcd B ) ) = b ) -> ( a gcd b ) = 1 ) )
94	84, 93	sylbid 150	. . . . . . . 8 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> ( a gcd b ) = 1 ) )
95	94	imp 124	. . . . . . 7 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> ( a gcd b ) = 1 )
96	28, 46, 95	3jca 1177	. . . . . 6 |- ( ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) /\ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) ) -> ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) )
97	96	ex 115	. . . . 5 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) ) )
98	97	reximdva 2579	. . . 4 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> ( E. b e. ZZ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) ) )
99	98	reximdva 2579	. . 3 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( E. a e. ZZ E. b e. ZZ ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) ) )
100	10, 99	biimtrrid 153	. 2 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( ( E. a e. ZZ A = ( a x. ( A gcd B ) ) /\ E. b e. ZZ B = ( b x. ( B gcd A ) ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) ) )
101	5, 9, 100	mp2and 433	1 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> E. a e. ZZ E. b e. ZZ ( A = ( M x. a ) /\ B = ( M x. b ) /\ ( a gcd b ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   E.wrex 2456   class class class wbr 4004  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812   x. cmul 7816   # cap 8538   / cdiv 8629   NN0cn0 9176   ZZcz 9253   gcd cgcd 11943

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-sup 6983  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944

This theorem is referenced by:  cncongr1  12103