Theorem divgcdcoprmex 12824

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 12691	. . . 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 1044	. 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 12691	. . . . 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 1044	. 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 2715	. . 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 9599	. . . . . . . . . . . 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 12687	. . . . . . . . . . . . . . 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 9572	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) e. CC )
16	15	3adant3 1044	. . . . . . . . . . . 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 8311	. . . . . . . . . 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 1026	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> M = ( A gcd B ) )
20	19	eqcomd 2240	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) = M )
21	20	oveq1d 6073	. . . . . . . . . . 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 2267	. . . . . . . . 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 2241	. . . . . . . . . 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 12694	. . . . . . . . . . . . . 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 1044	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( B gcd A ) = ( A gcd B ) )
34	33	oveq2d 6074	. . . . . . . . . . 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 9599	. . . . . . . . . . . 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 1044	. . . . . . . . . . . . 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 9572	. . . . . . . . . . 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 8311	. . . . . . . . . 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 6073	. . . . . . . . . 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 2271	. . . . . . . . 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 2289	. . . . . . 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 9599	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> A e. CC )
48	47	3ad2ant1 1045	. . . . . . . . . . . . 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 1024	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. ZZ )
52	1	3ad2ant2 1046	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. ZZ )
53	51, 52	gcdcld 12689	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
54	53	nn0cnd 9572	. . . . . . . . . . . . 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 12698	. . . . . . . . . . . . . . . . . 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 2458	. . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . . 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 9716	. . . . . . . . . . . . . 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 9606	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> 0 e. ZZ )
66		zapne 9669	. . . . . . . . . . . . . 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 9116	. . . . . . . . . . 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 9599	. . . . . . . . . . . . . . 15 |- ( B e. ZZ -> B e. CC )
72	71	adantr 276	. . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. CC )
73	72	3ad2ant2 1046	. . . . . . . . . . . . 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 9116	. . . . . . . . . . 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 1044	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ ) )
78		gcdcom 12694	. . . . . . . . . . . . . . 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 6074	. . . . . . . . . . . 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 2246	. . . . . . . . . . 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 1009	. . . . . . . . . . . . . 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 1044	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
88		divgcdcoprm0 12823	. . . . . . . . . . . 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 6067	. . . . . . . . . . . 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 2243	. . . . . . . . . . 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 1204	. . . . . 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 2646	. . . 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 2646	. . 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 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   x. cmul 8148   # cap 8872   / cdiv 8963   NN0cn0 9513   ZZcz 9594   gcd cgcd 12674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675

This theorem is referenced by:  cncongr1  12825