Theorem qredeq 12079

Description: Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.)

Assertion

Ref	Expression
qredeq	|- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) /\ ( M / N ) = ( P / Q ) ) -> ( M = P /\ N = Q ) )

Proof of Theorem qredeq

Step	Hyp	Ref	Expression
1		zcn 9247	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
2	1	adantr 276	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
3		nncn 8916	. . . . . . . . . 10 |- ( N e. NN -> N e. CC )
4	3	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
5		nnap0 8937	. . . . . . . . . 10 |- ( N e. NN -> N # 0 )
6	5	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N # 0 )
7	2, 4, 6	divclapd 8736	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. CC )
8	7	3adant3 1017	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M / N ) e. CC )
9	8	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( M / N ) e. CC )
10		zcn 9247	. . . . . . . . . 10 |- ( P e. ZZ -> P e. CC )
11	10	adantr 276	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> P e. CC )
12		nncn 8916	. . . . . . . . . 10 |- ( Q e. NN -> Q e. CC )
13	12	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q e. CC )
14		nnap0 8937	. . . . . . . . . 10 |- ( Q e. NN -> Q # 0 )
15	14	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q # 0 )
16	11, 13, 15	divclapd 8736	. . . . . . . 8 |- ( ( P e. ZZ /\ Q e. NN ) -> ( P / Q ) e. CC )
17	16	3adant3 1017	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( P / Q ) e. CC )
18	17	adantl 277	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( P / Q ) e. CC )
19	3	3ad2ant2 1019	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. CC )
20	19	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> N e. CC )
21	5	3ad2ant2 1019	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N # 0 )
22	21	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> N # 0 )
23	9, 18, 20, 22	mulcanapd 8607	. . . . 5 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( N x. ( M / N ) ) = ( N x. ( P / Q ) ) <-> ( M / N ) = ( P / Q ) ) )
24	2, 4, 6	divcanap2d 8738	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( N x. ( M / N ) ) = M )
25	24	3adant3 1017	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N x. ( M / N ) ) = M )
26	25	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N x. ( M / N ) ) = M )
27	26	eqeq1d 2186	. . . . 5 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( N x. ( M / N ) ) = ( N x. ( P / Q ) ) <-> M = ( N x. ( P / Q ) ) ) )
28	23, 27	bitr3d 190	. . . 4 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M / N ) = ( P / Q ) <-> M = ( N x. ( P / Q ) ) ) )
29	1	3ad2ant1 1018	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> M e. CC )
30	29	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> M e. CC )
31		mulcl 7929	. . . . . . 7 |- ( ( N e. CC /\ ( P / Q ) e. CC ) -> ( N x. ( P / Q ) ) e. CC )
32	19, 17, 31	syl2an 289	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N x. ( P / Q ) ) e. CC )
33	12	3ad2ant2 1019	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. CC )
34	33	adantl 277	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> Q e. CC )
35	14	3ad2ant2 1019	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q # 0 )
36	35	adantl 277	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> Q # 0 )
37	30, 32, 34, 36	mulcanap2d 8608	. . . . 5 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M x. Q ) = ( ( N x. ( P / Q ) ) x. Q ) <-> M = ( N x. ( P / Q ) ) ) )
38	20, 18, 34	mulassd 7971	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( N x. ( P / Q ) ) x. Q ) = ( N x. ( ( P / Q ) x. Q ) ) )
39	11, 13, 15	divcanap1d 8737	. . . . . . . . . 10 |- ( ( P e. ZZ /\ Q e. NN ) -> ( ( P / Q ) x. Q ) = P )
40	39	3adant3 1017	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( ( P / Q ) x. Q ) = P )
41	40	adantl 277	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( P / Q ) x. Q ) = P )
42	41	oveq2d 5885	. . . . . . 7 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N x. ( ( P / Q ) x. Q ) ) = ( N x. P ) )
43	38, 42	eqtrd 2210	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( N x. ( P / Q ) ) x. Q ) = ( N x. P ) )
44	43	eqeq2d 2189	. . . . 5 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M x. Q ) = ( ( N x. ( P / Q ) ) x. Q ) <-> ( M x. Q ) = ( N x. P ) ) )
45	37, 44	bitr3d 190	. . . 4 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( M = ( N x. ( P / Q ) ) <-> ( M x. Q ) = ( N x. P ) ) )
46	28, 45	bitrd 188	. . 3 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M / N ) = ( P / Q ) <-> ( M x. Q ) = ( N x. P ) ) )
47		nnz 9261	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
48	47	3ad2ant2 1019	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. ZZ )
49		simp2 998	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. NN )
50	48, 49	anim12i 338	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N e. ZZ /\ Q e. NN ) )
51	50	adantr 276	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N e. ZZ /\ Q e. NN ) )
52	48	adantr 276	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> N e. ZZ )
53		simpl1 1000	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> M e. ZZ )
54		nnz 9261	. . . . . . . . . . . 12 |- ( Q e. NN -> Q e. ZZ )
55	54	3ad2ant2 1019	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. ZZ )
56	55	adantl 277	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> Q e. ZZ )
57	52, 53, 56	3jca 1177	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N e. ZZ /\ M e. ZZ /\ Q e. ZZ ) )
58	57	adantr 276	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N e. ZZ /\ M e. ZZ /\ Q e. ZZ ) )
59		simp1 997	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. ZZ )
60		dvdsmul1 11804	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. ZZ ) -> N || ( N x. P ) )
61	48, 59, 60	syl2an 289	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> N || ( N x. P ) )
62	61	adantr 276	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N || ( N x. P ) )
63		simpr 110	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( M x. Q ) = ( N x. P ) )
64	62, 63	breqtrrd 4028	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N || ( M x. Q ) )
65		gcdcom 11957	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
66	47, 65	sylan 283	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
67	66	ancoms 268	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( M gcd N ) )
68	67	3adant3 1017	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = ( M gcd N ) )
69		simp3 999	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
70	68, 69	eqtrd 2210	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = 1 )
71	70	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N gcd M ) = 1 )
72	64, 71	jca 306	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N || ( M x. Q ) /\ ( N gcd M ) = 1 ) )
73		coprmdvds 12075	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ /\ Q e. ZZ ) -> ( ( N || ( M x. Q ) /\ ( N gcd M ) = 1 ) -> N || Q ) )
74	58, 72, 73	sylc 62	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N || Q )
75		dvdsle 11833	. . . . . . 7 |- ( ( N e. ZZ /\ Q e. NN ) -> ( N || Q -> N <_ Q ) )
76	51, 74, 75	sylc 62	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N <_ Q )
77		simp2 998	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. NN )
78	55, 77	anim12i 338	. . . . . . . . 9 |- ( ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) /\ ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( Q e. ZZ /\ N e. NN ) )
79	78	ancoms 268	. . . . . . . 8 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( Q e. ZZ /\ N e. NN ) )
80	79	adantr 276	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( Q e. ZZ /\ N e. NN ) )
81		simpr1 1003	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> P e. ZZ )
82	56, 81, 52	3jca 1177	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( Q e. ZZ /\ P e. ZZ /\ N e. ZZ ) )
83	82	adantr 276	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( Q e. ZZ /\ P e. ZZ /\ N e. ZZ ) )
84		simp1 997	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> M e. ZZ )
85		dvdsmul2 11805	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ Q e. ZZ ) -> Q || ( M x. Q ) )
86	84, 55, 85	syl2an 289	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> Q || ( M x. Q ) )
87	86	adantr 276	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> Q || ( M x. Q ) )
88	10	3ad2ant1 1018	. . . . . . . . . . . . 13 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. CC )
89		mulcom 7931	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ P e. CC ) -> ( N x. P ) = ( P x. N ) )
90	19, 88, 89	syl2an 289	. . . . . . . . . . . 12 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( N x. P ) = ( P x. N ) )
91	90	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N x. P ) = ( P x. N ) )
92	63, 91	eqtrd 2210	. . . . . . . . . 10 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( M x. Q ) = ( P x. N ) )
93	87, 92	breqtrd 4026	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> Q || ( P x. N ) )
94		gcdcom 11957	. . . . . . . . . . . . . 14 |- ( ( Q e. ZZ /\ P e. ZZ ) -> ( Q gcd P ) = ( P gcd Q ) )
95	54, 94	sylan 283	. . . . . . . . . . . . 13 |- ( ( Q e. NN /\ P e. ZZ ) -> ( Q gcd P ) = ( P gcd Q ) )
96	95	ancoms 268	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ Q e. NN ) -> ( Q gcd P ) = ( P gcd Q ) )
97	96	3adant3 1017	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( Q gcd P ) = ( P gcd Q ) )
98		simp3 999	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( P gcd Q ) = 1 )
99	97, 98	eqtrd 2210	. . . . . . . . . 10 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( Q gcd P ) = 1 )
100	99	ad2antlr 489	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( Q gcd P ) = 1 )
101	93, 100	jca 306	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( Q || ( P x. N ) /\ ( Q gcd P ) = 1 ) )
102		coprmdvds 12075	. . . . . . . 8 |- ( ( Q e. ZZ /\ P e. ZZ /\ N e. ZZ ) -> ( ( Q || ( P x. N ) /\ ( Q gcd P ) = 1 ) -> Q || N ) )
103	83, 101, 102	sylc 62	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> Q || N )
104		dvdsle 11833	. . . . . . 7 |- ( ( Q e. ZZ /\ N e. NN ) -> ( Q || N -> Q <_ N ) )
105	80, 103, 104	sylc 62	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> Q <_ N )
106		nnre 8915	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
107	106	3ad2ant2 1019	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. RR )
108	107	ad2antrr 488	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N e. RR )
109		nnre 8915	. . . . . . . . 9 |- ( Q e. NN -> Q e. RR )
110	109	3ad2ant2 1019	. . . . . . . 8 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. RR )
111	110	ad2antlr 489	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> Q e. RR )
112	108, 111	letri3d 8063	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N = Q <-> ( N <_ Q /\ Q <_ N ) ) )
113	76, 105, 112	mpbir2and 944	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> N = Q )
114		oveq2 5877	. . . . . . . . . 10 |- ( N = Q -> ( M x. N ) = ( M x. Q ) )
115	114	eqeq1d 2186	. . . . . . . . 9 |- ( N = Q -> ( ( M x. N ) = ( N x. P ) <-> ( M x. Q ) = ( N x. P ) ) )
116	115	anbi2d 464	. . . . . . . 8 |- ( N = Q -> ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. N ) = ( N x. P ) ) <-> ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) ) )
117		mulcom 7931	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ N e. CC ) -> ( M x. N ) = ( N x. M ) )
118	1, 3, 117	syl2an 289	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. NN ) -> ( M x. N ) = ( N x. M ) )
119	118	3adant3 1017	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M x. N ) = ( N x. M ) )
120	119	adantr 276	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( M x. N ) = ( N x. M ) )
121	120	eqeq1d 2186	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M x. N ) = ( N x. P ) <-> ( N x. M ) = ( N x. P ) ) )
122	88	adantl 277	. . . . . . . . . . 11 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> P e. CC )
123	30, 122, 20, 22	mulcanapd 8607	. . . . . . . . . 10 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( N x. M ) = ( N x. P ) <-> M = P ) )
124	121, 123	bitrd 188	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M x. N ) = ( N x. P ) <-> M = P ) )
125	124	biimpa 296	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. N ) = ( N x. P ) ) -> M = P )
126	116, 125	syl6bir 164	. . . . . . 7 |- ( N = Q -> ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> M = P ) )
127	126	com12 30	. . . . . 6 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N = Q -> M = P ) )
128	127	ancrd 326	. . . . 5 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( N = Q -> ( M = P /\ N = Q ) ) )
129	113, 128	mpd 13	. . . 4 |- ( ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) /\ ( M x. Q ) = ( N x. P ) ) -> ( M = P /\ N = Q ) )
130	129	ex 115	. . 3 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M x. Q ) = ( N x. P ) -> ( M = P /\ N = Q ) ) )
131	46, 130	sylbid 150	. 2 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) ) -> ( ( M / N ) = ( P / Q ) -> ( M = P /\ N = Q ) ) )
132	131	3impia 1200	1 |- ( ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) /\ ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) /\ ( M / N ) = ( P / Q ) ) -> ( M = P /\ N = Q ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   x. cmul 7807   <_ cle 7983   # cap 8528   / cdiv 8618   NNcn 8908   ZZcz 9242   || cdvds 11778   gcd cgcd 11926

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927

This theorem is referenced by:  qredeu  12080