Theorem divgcdcoprmex 12034

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 108	. . . . 5 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. ZZ )
2	1	anim2i 340	. . . 4 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A e. ZZ /\ B e. ZZ ) )
3		zeqzmulgcd 11903	. . . 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 1007	. 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 11903	. . . . 5 |- ( ( B e. ZZ /\ A e. ZZ ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
7	6	adantlr 469	. . . 4 |- ( ( ( B e. ZZ /\ B =/= 0 ) /\ A e. ZZ ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
8	7	ancoms 266	. . 3 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> E. b e. ZZ B = ( b x. ( B gcd A ) ) )
9	8	3adant3 1007	. 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 2635	. . 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 9196	. . . . . . . . . . . 12 |- ( a e. ZZ -> a e. CC )
12	11	adantl 275	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) -> a e. CC )
13		gcdcl 11899	. . . . . . . . . . . . . . 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 9169	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) e. CC )
16	15	3adant3 1007	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. CC )
17	16	adantr 274	. . . . . . . . . . 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 7920	. . . . . . . . . 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 989	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> M = ( A gcd B ) )
20	19	eqcomd 2171	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) = M )
21	20	oveq1d 5857	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( ( A gcd B ) x. a ) = ( M x. a ) )
22	21	adantr 274	. . . . . . . . . 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 2198	. . . . . . . . 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 480	. . . . . . . 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 2172	. . . . . . . . . 10 |- ( A = ( a x. ( A gcd B ) ) -> ( A = ( M x. a ) <-> ( a x. ( A gcd B ) ) = ( M x. a ) ) )
26	25	adantr 274	. . . . . . . . 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 275	. . . . . . . 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 166	. . . . . . 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 109	. . . . . . . 8 |- ( ( A = ( a x. ( A gcd B ) ) /\ B = ( b x. ( B gcd A ) ) ) -> B = ( b x. ( B gcd A ) ) )
30	2	ancomd 265	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( B e. ZZ /\ A e. ZZ ) )
31		gcdcom 11906	. . . . . . . . . . . . . 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 1007	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( B gcd A ) = ( A gcd B ) )
34	33	oveq2d 5858	. . . . . . . . . . 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 274	. . . . . . . . . 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 9196	. . . . . . . . . . . 12 |- ( b e. ZZ -> b e. CC )
37	36	adantl 275	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> b e. CC )
38	14	3adant3 1007	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
39	38	adantr 274	. . . . . . . . . . . 12 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( A gcd B ) e. NN0 )
40	39	nn0cnd 9169	. . . . . . . . . . 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 7920	. . . . . . . . . 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 274	. . . . . . . . . . 11 |- ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ b e. ZZ ) -> ( A gcd B ) = M )
43	42	oveq1d 5857	. . . . . . . . . 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 2202	. . . . . . . . 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 469	. . . . . . . 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 2221	. . . . . . 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 9196	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> A e. CC )
48	47	3ad2ant1 1008	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. CC )
49	48	ad2antrr 480	. . . . . . . . . . . 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 274	. . . . . . . . . . . 12 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> a e. CC )
51		simp1 987	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> A e. ZZ )
52	1	3ad2ant2 1009	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. ZZ )
53	51, 52	gcdcld 11901	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. NN0 )
54	53	nn0cnd 9169	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) e. CC )
55	54	ad2antrr 480	. . . . . . . . . . . 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 11910	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
57		simpr 109	. . . . . . . . . . . . . . . . . 18 |- ( ( A = 0 /\ B = 0 ) -> B = 0 )
58	56, 57	syl6bi 162	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 -> B = 0 ) )
59	58	necon3d 2380	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( B =/= 0 -> ( A gcd B ) =/= 0 ) )
60	59	impr 377	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A gcd B ) =/= 0 )
61	60	3adant3 1007	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A gcd B ) =/= 0 )
62	61	ad2antrr 480	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 9311	. . . . . . . . . . . . . 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 9203	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) /\ a e. ZZ ) /\ b e. ZZ ) -> 0 e. ZZ )
66		zapne 9265	. . . . . . . . . . . . . 14 |- ( ( ( A gcd B ) e. ZZ /\ 0 e. ZZ ) -> ( ( A gcd B ) # 0 <-> ( A gcd B ) =/= 0 ) )
67	64, 65, 66	syl2anc 409	. . . . . . . . . . . . 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 166	. . . . . . . . . . . 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 8721	. . . . . . . . . . 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 140	. . . . . . . . . 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 9196	. . . . . . . . . . . . . . 15 |- ( B e. ZZ -> B e. CC )
72	71	adantr 274	. . . . . . . . . . . . . 14 |- ( ( B e. ZZ /\ B =/= 0 ) -> B e. CC )
73	72	3ad2ant2 1009	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> B e. CC )
74	73	ad2antrr 480	. . . . . . . . . . . 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 275	. . . . . . . . . . . 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 8721	. . . . . . . . . . 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 1007	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ ) )
78		gcdcom 11906	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 5858	. . . . . . . . . . . 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 2177	. . . . . . . . . . 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 188	. . . . . . . . . 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 465	. . . . . . . . 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 972	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) <-> ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) )
86	85	biimpri 132	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
87	86	3adant3 1007	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( B e. ZZ /\ B =/= 0 ) /\ M = ( A gcd B ) ) -> ( A e. ZZ /\ B e. ZZ /\ B =/= 0 ) )
88		divgcdcoprm0 12033	. . . . . . . . . . . 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 5851	. . . . . . . . . . . 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 2174	. . . . . . . . . . 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 154	. . . . . . . . . 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 480	. . . . . . . . 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 149	. . . . . . . 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 123	. . . . . . 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 1167	. . . . . 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 114	. . . . 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 2568	. . . 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 2568	. . 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	syl5bir 152	. 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 430	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   x. cmul 7758   # cap 8479   / cdiv 8568   NN0cn0 9114   ZZcz 9191   gcd cgcd 11875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876

This theorem is referenced by:  cncongr1  12035