Theorem qredeu 10985

Description: Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)

Assertion

Ref	Expression
qredeu	|- ( A e. QQ -> E! x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )

Distinct variable group:   x, A

Proof of Theorem qredeu

Dummy variables n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnz 8705	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
2		gcddvds 10861	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( ( z gcd n ) || z /\ ( z gcd n ) || n ) )
3	2	simpld 110	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || z )
4	1, 3	sylan2 280	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || z )
5		gcdcl 10864	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) e. NN0 )
6	1, 5	sylan2 280	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN0 )
7	6	nn0zd 8802	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. ZZ )
8		simpl 107	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. ZZ )
9	1	adantl 271	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. ZZ )
10		nnne0 8388	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
11	10	neneqd 2272	. . . . . . . . . . . . . 14 |- ( n e. NN -> -. n = 0 )
12	11	intnand 876	. . . . . . . . . . . . 13 |- ( n e. NN -> -. ( z = 0 /\ n = 0 ) )
13	12	adantl 271	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> -. ( z = 0 /\ n = 0 ) )
14		gcdn0cl 10860	. . . . . . . . . . . 12 |- ( ( ( z e. ZZ /\ n e. ZZ ) /\ -. ( z = 0 /\ n = 0 ) ) -> ( z gcd n ) e. NN )
15	8, 9, 13, 14	syl21anc 1171	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN )
16	15	nnne0d 8404	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) =/= 0 )
17		dvdsval2 10705	. . . . . . . . . 10 |- ( ( ( z gcd n ) e. ZZ /\ ( z gcd n ) =/= 0 /\ z e. ZZ ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
18	7, 16, 8, 17	syl3anc 1172	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
19	4, 18	mpbid 145	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( z / ( z gcd n ) ) e. ZZ )
20	19	3adant3 961	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. ZZ )
21	2	simprd 112	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || n )
22	1, 21	sylan2 280	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || n )
23		dvdsval2 10705	. . . . . . . . . . . 12 |- ( ( ( z gcd n ) e. ZZ /\ ( z gcd n ) =/= 0 /\ n e. ZZ ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
24	7, 16, 9, 23	syl3anc 1172	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
25	22, 24	mpbid 145	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( n / ( z gcd n ) ) e. ZZ )
26		nnre 8367	. . . . . . . . . . . 12 |- ( n e. NN -> n e. RR )
27	26	adantl 271	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. RR )
28	6	nn0red 8663	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. RR )
29		nngt0 8385	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
30	29	adantl 271	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < n )
31	15	nngt0d 8403	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( z gcd n ) )
32	27, 28, 30, 31	divgt0d 8334	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( n / ( z gcd n ) ) )
33	25, 32	jca 300	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
34	33	3adant3 961	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
35		elnnz 8696	. . . . . . . 8 |- ( ( n / ( z gcd n ) ) e. NN <-> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
36	34, 35	sylibr 132	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. NN )
37		opelxpi 4444	. . . . . . 7 |- ( ( ( z / ( z gcd n ) ) e. ZZ /\ ( n / ( z gcd n ) ) e. NN ) -> <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) )
38	20, 36, 37	syl2anc 403	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) )
39		fveq2 5270	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 1st ` x ) = ( 1st ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
40		simp1 941	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> z e. ZZ )
41	15	3adant3 961	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z gcd n ) e. NN )
42		znq 9044	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ ( z gcd n ) e. NN ) -> ( z / ( z gcd n ) ) e. QQ )
43	40, 41, 42	syl2anc 403	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. QQ )
44	9	3adant3 961	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> n e. ZZ )
45		znq 9044	. . . . . . . . . . . 12 |- ( ( n e. ZZ /\ ( z gcd n ) e. NN ) -> ( n / ( z gcd n ) ) e. QQ )
46	44, 41, 45	syl2anc 403	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. QQ )
47		op1stg 5880	. . . . . . . . . . 11 |- ( ( ( z / ( z gcd n ) ) e. QQ /\ ( n / ( z gcd n ) ) e. QQ ) -> ( 1st ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) = ( z / ( z gcd n ) ) )
48	43, 46, 47	syl2anc 403	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( 1st ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) = ( z / ( z gcd n ) ) )
49	39, 48	sylan9eqr 2139	. . . . . . . . 9 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( 1st ` x ) = ( z / ( z gcd n ) ) )
50		fveq2 5270	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 2nd ` x ) = ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
51		op2ndg 5881	. . . . . . . . . . 11 |- ( ( ( z / ( z gcd n ) ) e. QQ /\ ( n / ( z gcd n ) ) e. QQ ) -> ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) = ( n / ( z gcd n ) ) )
52	43, 46, 51	syl2anc 403	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) = ( n / ( z gcd n ) ) )
53	50, 52	sylan9eqr 2139	. . . . . . . . 9 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( 2nd ` x ) = ( n / ( z gcd n ) ) )
54	49, 53	oveq12d 5633	. . . . . . . 8 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) )
55	54	eqeq1d 2093	. . . . . . 7 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 ) )
56	49, 53	oveq12d 5633	. . . . . . . 8 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
57	56	eqeq2d 2096	. . . . . . 7 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) ) )
58	55, 57	anbi12d 457	. . . . . 6 |- ( ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) /\ x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) -> ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) <-> ( ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 /\ A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) ) ) )
59	19, 25	gcdcld 10866	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. NN0 )
60	59	nn0cnd 8664	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. CC )
61		1cnd 7451	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> 1 e. CC )
62	6	nn0cnd 8664	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. CC )
63	15	nnap0d 8405	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) # 0 )
64	62	mulid1d 7452	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. 1 ) = ( z gcd n ) )
65		zcn 8691	. . . . . . . . . . . . 13 |- ( z e. ZZ -> z e. CC )
66	65	adantr 270	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. CC )
67	66, 62, 63	divcanap2d 8200	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) = z )
68		nncn 8368	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. CC )
69	68	adantl 271	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. CC )
70	69, 62, 63	divcanap2d 8200	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) = n )
71	67, 70	oveq12d 5633	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) gcd ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) ) = ( z gcd n ) )
72		mulgcd 10911	. . . . . . . . . . 11 |- ( ( ( z gcd n ) e. NN0 /\ ( z / ( z gcd n ) ) e. ZZ /\ ( n / ( z gcd n ) ) e. ZZ ) -> ( ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) gcd ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) ) = ( ( z gcd n ) x. ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) ) )
73	6, 19, 25, 72	syl3anc 1172	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) gcd ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) ) = ( ( z gcd n ) x. ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) ) )
74	64, 71, 73	3eqtr2rd 2124	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) ) = ( ( z gcd n ) x. 1 ) )
75	60, 61, 62, 63, 74	mulcanapad 8074	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
76	75	3adant3 961	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
77		nnap0 8389	. . . . . . . . . . 11 |- ( n e. NN -> n # 0 )
78	77	adantl 271	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> n # 0 )
79	66, 69, 62, 78, 63	divcanap7d 8226	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) = ( z / n ) )
80	79	eqeq2d 2096	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) <-> A = ( z / n ) ) )
81	80	biimp3ar 1280	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
82	76, 81	jca 300	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 /\ A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) ) )
83	38, 58, 82	rspcedvd 2721	. . . . 5 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )
84		elxp6 5899	. . . . . . 7 |- ( x e. ( ZZ X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) )
85		elxp6 5899	. . . . . . 7 |- ( y e. ( ZZ X. NN ) <-> ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) )
86		simprl 498	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. ZZ )
87	86	ad2antrr 472	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( 1st ` x ) e. ZZ )
88		simprr 499	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. NN )
89	88	ad2antrr 472	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( 2nd ` x ) e. NN )
90		simprll 504	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 )
91		simprl 498	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 1st ` y ) e. ZZ )
92	91	ad2antlr 473	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( 1st ` y ) e. ZZ )
93		simprr 499	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 2nd ` y ) e. NN )
94	93	ad2antlr 473	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( 2nd ` y ) e. NN )
95		simprrl 506	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 )
96		simprlr 505	. . . . . . . . . . . 12 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> A = ( ( 1st ` x ) / ( 2nd ` x ) ) )
97		simprrr 507	. . . . . . . . . . . 12 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> A = ( ( 1st ` y ) / ( 2nd ` y ) ) )
98	96, 97	eqtr3d 2119	. . . . . . . . . . 11 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
99		qredeq 10984	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN /\ ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 ) /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN /\ ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) /\ ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) ) -> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
100	87, 89, 90, 92, 94, 95, 98, 99	syl331anc 1197	. . . . . . . . . 10 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
101		vex 2618	. . . . . . . . . . . 12 |- x e. _V
102		1stexg 5897	. . . . . . . . . . . 12 |- ( x e. _V -> ( 1st ` x ) e. _V )
103	101, 102	ax-mp 7	. . . . . . . . . . 11 |- ( 1st ` x ) e. _V
104		2ndexg 5898	. . . . . . . . . . . 12 |- ( x e. _V -> ( 2nd ` x ) e. _V )
105	101, 104	ax-mp 7	. . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
106	103, 105	opth 4040	. . . . . . . . . 10 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. ( 1st ` y ) , ( 2nd ` y ) >. <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
107	100, 106	sylibr 132	. . . . . . . . 9 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. ( 1st ` y ) , ( 2nd ` y ) >. )
108		simplll 500	. . . . . . . . 9 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
109		simplrl 502	. . . . . . . . 9 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
110	107, 108, 109	3eqtr4d 2127	. . . . . . . 8 |- ( ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) /\ ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) ) -> x = y )
111	110	ex 113	. . . . . . 7 |- ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) /\ ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) ) -> ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) )
112	84, 85, 111	syl2anb 285	. . . . . 6 |- ( ( x e. ( ZZ X. NN ) /\ y e. ( ZZ X. NN ) ) -> ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) )
113	112	rgen2a 2425	. . . . 5 |- A. x e. ( ZZ X. NN ) A. y e. ( ZZ X. NN ) ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y )
114	83, 113	jctir 306	. . . 4 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ A. x e. ( ZZ X. NN ) A. y e. ( ZZ X. NN ) ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) ) )
115	114	3expia 1143	. . 3 |- ( ( z e. ZZ /\ n e. NN ) -> ( A = ( z / n ) -> ( E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ A. x e. ( ZZ X. NN ) A. y e. ( ZZ X. NN ) ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) ) ) )
116	115	rexlimivv 2490	. 2 |- ( E. z e. ZZ E. n e. NN A = ( z / n ) -> ( E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ A. x e. ( ZZ X. NN ) A. y e. ( ZZ X. NN ) ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) ) )
117		elq 9042	. 2 |- ( A e. QQ <-> E. z e. ZZ E. n e. NN A = ( z / n ) )
118		fveq2 5270	. . . . . 6 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
119		fveq2 5270	. . . . . 6 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
120	118, 119	oveq12d 5633	. . . . 5 |- ( x = y -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( 1st ` y ) gcd ( 2nd ` y ) ) )
121	120	eqeq1d 2093	. . . 4 |- ( x = y -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) )
122	118, 119	oveq12d 5633	. . . . 5 |- ( x = y -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
123	122	eqeq2d 2096	. . . 4 |- ( x = y -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) )
124	121, 123	anbi12d 457	. . 3 |- ( x = y -> ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) <-> ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) )
125	124	reu4 2800	. 2 |- ( E! x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) <-> ( E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ A. x e. ( ZZ X. NN ) A. y e. ( ZZ X. NN ) ( ( ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) /\ ( ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 /\ A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) ) -> x = y ) ) )
126	116, 117, 125	3imtr4i 199	1 |- ( A e. QQ -> E! x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 922   = wceq 1287   e. wcel 1436   =/= wne 2251   A.wral 2355   E.wrex 2356   E!wreu 2357   _Vcvv 2615   <.cop 3434   class class class wbr 3822   X. cxp 4411   ` cfv 4983  (class class class)co 5615   1stc1st 5868   2ndc2nd 5869   CCcc 7295   RRcr 7296   0cc0 7297   1c1 7298   x. cmul 7302   < clt 7469   # cap 8002   / cdiv 8081   NNcn 8360   NN0cn0 8609   ZZcz 8686   QQcq 9039   || cdvds 10702   gcd cgcd 10844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-precex 7402  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-apti 7407  ax-pre-ltadd 7408  ax-pre-mulgt0 7409  ax-pre-mulext 7410  ax-arch 7411  ax-caucvg 7412

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-frec 6112  df-sup 6626  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-reap 7996  df-ap 8003  df-div 8082  df-inn 8361  df-2 8419  df-3 8420  df-4 8421  df-n0 8610  df-z 8687  df-uz 8955  df-q 9040  df-rp 9070  df-fz 9360  df-fzo 9485  df-fl 9608  df-mod 9661  df-iseq 9783  df-iexp 9857  df-cj 10175  df-re 10176  df-im 10177  df-rsqrt 10330  df-abs 10331  df-dvds 10703  df-gcd 10845

This theorem is referenced by:  qnumdencl  11071  qnumdenbi  11076