Theorem qredeq 12234

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 9322	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
2	1	adantr 276	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
3		nncn 8990	. . . . . . . . . 10 |- ( N e. NN -> N e. CC )
4	3	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
5		nnap0 9011	. . . . . . . . . 10 |- ( N e. NN -> N # 0 )
6	5	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N # 0 )
7	2, 4, 6	divclapd 8809	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. CC )
8	7	3adant3 1019	. . . . . . 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 9322	. . . . . . . . . 10 |- ( P e. ZZ -> P e. CC )
11	10	adantr 276	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> P e. CC )
12		nncn 8990	. . . . . . . . . 10 |- ( Q e. NN -> Q e. CC )
13	12	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q e. CC )
14		nnap0 9011	. . . . . . . . . 10 |- ( Q e. NN -> Q # 0 )
15	14	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q # 0 )
16	11, 13, 15	divclapd 8809	. . . . . . . 8 |- ( ( P e. ZZ /\ Q e. NN ) -> ( P / Q ) e. CC )
17	16	3adant3 1019	. . . . . . 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 1021	. . . . . . 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 1021	. . . . . . 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 8680	. . . . 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 8811	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( N x. ( M / N ) ) = M )
25	24	3adant3 1019	. . . . . . 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 2202	. . . . 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 1020	. . . . . . 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 7999	. . . . . . 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 1021	. . . . . . 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 1021	. . . . . . 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 8681	. . . . 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 8043	. . . . . . 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 8810	. . . . . . . . . 10 |- ( ( P e. ZZ /\ Q e. NN ) -> ( ( P / Q ) x. Q ) = P )
40	39	3adant3 1019	. . . . . . . . 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 5934	. . . . . . 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 2226	. . . . . 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 2205	. . . . 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 9336	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
48	47	3ad2ant2 1021	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. ZZ )
49		simp2 1000	. . . . . . . . 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 1002	. . . . . . . . . 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 9336	. . . . . . . . . . . 12 |- ( Q e. NN -> Q e. ZZ )
55	54	3ad2ant2 1021	. . . . . . . . . . 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 1179	. . . . . . . . 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 999	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. ZZ )
60		dvdsmul1 11956	. . . . . . . . . . . 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 4057	. . . . . . . . 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 12110	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = ( M gcd N ) )
69		simp3 1001	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
70	68, 69	eqtrd 2226	. . . . . . . . . 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 12230	. . . . . . . 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 11986	. . . . . . 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 1000	. . . . . . . . . 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 1005	. . . . . . . . . 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 1179	. . . . . . . . 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 999	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> M e. ZZ )
85		dvdsmul2 11957	. . . . . . . . . . . 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 1020	. . . . . . . . . . . . 13 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. CC )
89		mulcom 8001	. . . . . . . . . . . . 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 2226	. . . . . . . . . 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 4055	. . . . . . . . 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 12110	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( Q gcd P ) = ( P gcd Q ) )
98		simp3 1001	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( P gcd Q ) = 1 )
99	97, 98	eqtrd 2226	. . . . . . . . . 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 12230	. . . . . . . 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 11986	. . . . . . 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 8989	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
107	106	3ad2ant2 1021	. . . . . . . 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 8989	. . . . . . . . 9 |- ( Q e. NN -> Q e. RR )
110	109	3ad2ant2 1021	. . . . . . . 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 8135	. . . . . 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 946	. . . . 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 5926	. . . . . . . . . 10 |- ( N = Q -> ( M x. N ) = ( M x. Q ) )
115	114	eqeq1d 2202	. . . . . . . . 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 8001	. . . . . . . . . . . . . 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 1019	. . . . . . . . . . . 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 2202	. . . . . . . . . 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 8680	. . . . . . . . . 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	biimtrrdi 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 1202	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 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   x. cmul 7877   <_ cle 8055   # cap 8600   / cdiv 8691   NNcn 8982   ZZcz 9317   || cdvds 11930   gcd cgcd 12079

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080

This theorem is referenced by:  qredeu  12235