Theorem divgcdcoprmex 12270

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 12137	. . . 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 1019	. 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 12137	. . . . 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 1019	. 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 2667	. . 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 9331	. . . . . . . . . . . 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 12133	. . . . . . . . . . . . . . 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 9304	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) e. CC )
16	15	3adant3 1019	. . . . . . . . . . . 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 8048	. . . . . . . . . 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 1001	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> M = ( A gcd B ) )
20	19	eqcomd 2202	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) = M )
21	20	oveq1d 5937	. . . . . . . . . . 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 2229	. . . . . . . . 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 2203	. . . . . . . . . 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 12140	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( B gcd A ) = ( A gcd B ) )
34	33	oveq2d 5938	. . . . . . . . . . 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 9331	. . . . . . . . . . . 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 1019	. . . . . . . . . . . . 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 9304	. . . . . . . . . . 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 8048	. . . . . . . . . 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 5937	. . . . . . . . . 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 2233	. . . . . . . . 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 2251	. . . . . . 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 9331	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> A e. CC )
48	47	3ad2ant1 1020	. . . . . . . . . . . . 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 999	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. ZZ )
52	1	3ad2ant2 1021	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. ZZ )
53	51, 52	gcdcld 12135	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
54	53	nn0cnd 9304	. . . . . . . . . . . . 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 12144	. . . . . . . . . . . . . . . . . 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 2411	. . . . . . . . . . . . . . . 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 1019	. . . . . . . . . . . . . 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 9446	. . . . . . . . . . . . . 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 9338	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> 0 e. ZZ )
66		zapne 9400	. . . . . . . . . . . . . 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 8852	. . . . . . . . . . 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 9331	. . . . . . . . . . . . . . 15 |- ( B e. ZZ -> B e. CC )
72	71	adantr 276	. . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. CC )
73	72	3ad2ant2 1021	. . . . . . . . . . . . 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 8852	. . . . . . . . . . 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 1019	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ ) )
78		gcdcom 12140	. . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . 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 2208	. . . . . . . . . . 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 984	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
88		divgcdcoprm0 12269	. . . . . . . . . . . 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 5931	. . . . . . . . . . . 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 2205	. . . . . . . . . . 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 1179	. . . . . 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 2599	. . . 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 2599	. . 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 980   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   x. cmul 7884   # cap 8608   / cdiv 8699   NN0cn0 9249   ZZcz 9326   gcd cgcd 12120

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121

This theorem is referenced by:  cncongr1  12271