Theorem qredeq 11766

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 9052	. . . . . . . . . 10 |- ( M e. ZZ -> M e. CC )
2	1	adantr 274	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> M e. CC )
3		nncn 8721	. . . . . . . . . 10 |- ( N e. NN -> N e. CC )
4	3	adantl 275	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N e. CC )
5		nnap0 8742	. . . . . . . . . 10 |- ( N e. NN -> N # 0 )
6	5	adantl 275	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN ) -> N # 0 )
7	2, 4, 6	divclapd 8543	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. CC )
8	7	3adant3 1001	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M / N ) e. CC )
9	8	adantr 274	. . . . . 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 9052	. . . . . . . . . 10 |- ( P e. ZZ -> P e. CC )
11	10	adantr 274	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> P e. CC )
12		nncn 8721	. . . . . . . . . 10 |- ( Q e. NN -> Q e. CC )
13	12	adantl 275	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q e. CC )
14		nnap0 8742	. . . . . . . . . 10 |- ( Q e. NN -> Q # 0 )
15	14	adantl 275	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN ) -> Q # 0 )
16	11, 13, 15	divclapd 8543	. . . . . . . 8 |- ( ( P e. ZZ /\ Q e. NN ) -> ( P / Q ) e. CC )
17	16	3adant3 1001	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( P / Q ) e. CC )
18	17	adantl 275	. . . . . 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 1003	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. CC )
20	19	adantr 274	. . . . . 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 1003	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N # 0 )
22	21	adantr 274	. . . . . 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 8415	. . . . 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 8545	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( N x. ( M / N ) ) = M )
25	24	3adant3 1001	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N x. ( M / N ) ) = M )
26	25	adantr 274	. . . . . 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 2146	. . . . 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 189	. . . 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 1002	. . . . . . 7 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> M e. CC )
30	29	adantr 274	. . . . . 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 7740	. . . . . . 7 |- ( ( N e. CC /\ ( P / Q ) e. CC ) -> ( N x. ( P / Q ) ) e. CC )
32	19, 17, 31	syl2an 287	. . . . . 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 1003	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. CC )
34	33	adantl 275	. . . . . 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 1003	. . . . . . 7 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q # 0 )
36	35	adantl 275	. . . . . 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 8416	. . . . 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 7782	. . . . . . 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 8544	. . . . . . . . . 10 |- ( ( P e. ZZ /\ Q e. NN ) -> ( ( P / Q ) x. Q ) = P )
40	39	3adant3 1001	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( ( P / Q ) x. Q ) = P )
41	40	adantl 275	. . . . . . . 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 5783	. . . . . . 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 2170	. . . . . 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 2149	. . . . 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 189	. . . 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 187	. . 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 9066	. . . . . . . . . 10 |- ( N e. NN -> N e. ZZ )
48	47	3ad2ant2 1003	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. ZZ )
49		simp2 982	. . . . . . . . 9 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. NN )
50	48, 49	anim12i 336	. . . . . . . 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 274	. . . . . . 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 274	. . . . . . . . . 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 984	. . . . . . . . . 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 9066	. . . . . . . . . . . 12 |- ( Q e. NN -> Q e. ZZ )
55	54	3ad2ant2 1003	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. ZZ )
56	55	adantl 275	. . . . . . . . . 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 1161	. . . . . . . . 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 274	. . . . . . . 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 981	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. ZZ )
60		dvdsmul1 11504	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ P e. ZZ ) -> N || ( N x. P ) )
61	48, 59, 60	syl2an 287	. . . . . . . . . . 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 274	. . . . . . . . . 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 109	. . . . . . . . . 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 3951	. . . . . . . . 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 11651	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
66	47, 65	sylan 281	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ M e. ZZ ) -> ( N gcd M ) = ( M gcd N ) )
67	66	ancoms 266	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN ) -> ( N gcd M ) = ( M gcd N ) )
68	67	3adant3 1001	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = ( M gcd N ) )
69		simp3 983	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M gcd N ) = 1 )
70	68, 69	eqtrd 2170	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( N gcd M ) = 1 )
71	70	ad2antrr 479	. . . . . . . . 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 304	. . . . . . . 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 11762	. . . . . . . 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 11531	. . . . . . 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 982	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. NN )
78	55, 77	anim12i 336	. . . . . . . . 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 266	. . . . . . . 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 274	. . . . . . 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 987	. . . . . . . . . 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 1161	. . . . . . . . 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 274	. . . . . . . 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 981	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> M e. ZZ )
85		dvdsmul2 11505	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ Q e. ZZ ) -> Q || ( M x. Q ) )
86	84, 55, 85	syl2an 287	. . . . . . . . . . 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 274	. . . . . . . . . 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 1002	. . . . . . . . . . . . 13 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> P e. CC )
89		mulcom 7742	. . . . . . . . . . . . 13 |- ( ( N e. CC /\ P e. CC ) -> ( N x. P ) = ( P x. N ) )
90	19, 88, 89	syl2an 287	. . . . . . . . . . . 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 274	. . . . . . . . . . 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 2170	. . . . . . . . . 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 3949	. . . . . . . . 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 11651	. . . . . . . . . . . . . 14 |- ( ( Q e. ZZ /\ P e. ZZ ) -> ( Q gcd P ) = ( P gcd Q ) )
95	54, 94	sylan 281	. . . . . . . . . . . . 13 |- ( ( Q e. NN /\ P e. ZZ ) -> ( Q gcd P ) = ( P gcd Q ) )
96	95	ancoms 266	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ Q e. NN ) -> ( Q gcd P ) = ( P gcd Q ) )
97	96	3adant3 1001	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( Q gcd P ) = ( P gcd Q ) )
98		simp3 983	. . . . . . . . . . 11 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( P gcd Q ) = 1 )
99	97, 98	eqtrd 2170	. . . . . . . . . 10 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> ( Q gcd P ) = 1 )
100	99	ad2antlr 480	. . . . . . . . 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 304	. . . . . . . 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 11762	. . . . . . . 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 11531	. . . . . . 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 8720	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
107	106	3ad2ant2 1003	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> N e. RR )
108	107	ad2antrr 479	. . . . . . 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 8720	. . . . . . . . 9 |- ( Q e. NN -> Q e. RR )
110	109	3ad2ant2 1003	. . . . . . . 8 |- ( ( P e. ZZ /\ Q e. NN /\ ( P gcd Q ) = 1 ) -> Q e. RR )
111	110	ad2antlr 480	. . . . . . 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 7872	. . . . . 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 928	. . . . 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 5775	. . . . . . . . . 10 |- ( N = Q -> ( M x. N ) = ( M x. Q ) )
115	114	eqeq1d 2146	. . . . . . . . 9 |- ( N = Q -> ( ( M x. N ) = ( N x. P ) <-> ( M x. Q ) = ( N x. P ) ) )
116	115	anbi2d 459	. . . . . . . 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 7742	. . . . . . . . . . . . . 14 |- ( ( M e. CC /\ N e. CC ) -> ( M x. N ) = ( N x. M ) )
118	1, 3, 117	syl2an 287	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ N e. NN ) -> ( M x. N ) = ( N x. M ) )
119	118	3adant3 1001	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ N e. NN /\ ( M gcd N ) = 1 ) -> ( M x. N ) = ( N x. M ) )
120	119	adantr 274	. . . . . . . . . . 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 2146	. . . . . . . . . 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 275	. . . . . . . . . . 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 8415	. . . . . . . . . 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 187	. . . . . . . . 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 294	. . . . . . . 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 163	. . . . . . 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 324	. . . . 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 114	. . 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 149	. 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 1178	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 103   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   x. cmul 7618   <_ cle 7794   # cap 8336   / cdiv 8425   NNcn 8713   ZZcz 9047   || cdvds 11482   gcd cgcd 11624

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625

This theorem is referenced by:  qredeu  11767