Theorem qredeu 12290

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 9362	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
2		gcddvds 12155	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( ( z gcd n ) || z /\ ( z gcd n ) || n ) )
3	2	simpld 112	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || z )
4	1, 3	sylan2 286	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || z )
5		gcdcl 12158	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) e. NN0 )
6	1, 5	sylan2 286	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN0 )
7	6	nn0zd 9463	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. ZZ )
8		simpl 109	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. ZZ )
9	1	adantl 277	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. ZZ )
10		nnne0 9035	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
11	10	neneqd 2388	. . . . . . . . . . . . . 14 |- ( n e. NN -> -. n = 0 )
12	11	intnand 932	. . . . . . . . . . . . 13 |- ( n e. NN -> -. ( z = 0 /\ n = 0 ) )
13	12	adantl 277	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> -. ( z = 0 /\ n = 0 ) )
14		gcdn0cl 12154	. . . . . . . . . . . 12 |- ( ( ( z e. ZZ /\ n e. ZZ ) /\ -. ( z = 0 /\ n = 0 ) ) -> ( z gcd n ) e. NN )
15	8, 9, 13, 14	syl21anc 1248	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN )
16	15	nnne0d 9052	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) =/= 0 )
17		dvdsval2 11972	. . . . . . . . . 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 1249	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
19	4, 18	mpbid 147	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( z / ( z gcd n ) ) e. ZZ )
20	19	3adant3 1019	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. ZZ )
21	2	simprd 114	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || n )
22	1, 21	sylan2 286	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || n )
23		dvdsval2 11972	. . . . . . . . . . . 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 1249	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
25	22, 24	mpbid 147	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( n / ( z gcd n ) ) e. ZZ )
26		nnre 9014	. . . . . . . . . . . 12 |- ( n e. NN -> n e. RR )
27	26	adantl 277	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. RR )
28	6	nn0red 9320	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. RR )
29		nngt0 9032	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
30	29	adantl 277	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < n )
31	15	nngt0d 9051	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( z gcd n ) )
32	27, 28, 30, 31	divgt0d 8979	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( n / ( z gcd n ) ) )
33	25, 32	jca 306	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
34	33	3adant3 1019	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
35		elnnz 9353	. . . . . . . 8 |- ( ( n / ( z gcd n ) ) e. NN <-> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
36	34, 35	sylibr 134	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. NN )
37		opelxpi 4696	. . . . . . 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 411	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) )
39		fveq2 5561	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 1st ` x ) = ( 1st ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
40		simp1 999	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> z e. ZZ )
41	15	3adant3 1019	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z gcd n ) e. NN )
42		znq 9715	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ ( z gcd n ) e. NN ) -> ( z / ( z gcd n ) ) e. QQ )
43	40, 41, 42	syl2anc 411	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. QQ )
44	9	3adant3 1019	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> n e. ZZ )
45		znq 9715	. . . . . . . . . . . 12 |- ( ( n e. ZZ /\ ( z gcd n ) e. NN ) -> ( n / ( z gcd n ) ) e. QQ )
46	44, 41, 45	syl2anc 411	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. QQ )
47		op1stg 6217	. . . . . . . . . . 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 411	. . . . . . . . . 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 2251	. . . . . . . . 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 5561	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 2nd ` x ) = ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
51		op2ndg 6218	. . . . . . . . . . 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 411	. . . . . . . . . 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 2251	. . . . . . . . 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 5943	. . . . . . . 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 2205	. . . . . . 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 5943	. . . . . . . 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 2208	. . . . . . 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 473	. . . . . 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 12160	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. NN0 )
60	59	nn0cnd 9321	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. CC )
61		1cnd 8059	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> 1 e. CC )
62	6	nn0cnd 9321	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. CC )
63	15	nnap0d 9053	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) # 0 )
64	62	mulridd 8060	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. 1 ) = ( z gcd n ) )
65		zcn 9348	. . . . . . . . . . . . 13 |- ( z e. ZZ -> z e. CC )
66	65	adantr 276	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. CC )
67	66, 62, 63	divcanap2d 8836	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) = z )
68		nncn 9015	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. CC )
69	68	adantl 277	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. CC )
70	69, 62, 63	divcanap2d 8836	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) = n )
71	67, 70	oveq12d 5943	. . . . . . . . . 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 12208	. . . . . . . . . . 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 1249	. . . . . . . . . 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 2236	. . . . . . . . 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 8707	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
76	75	3adant3 1019	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
77		nnap0 9036	. . . . . . . . . . 11 |- ( n e. NN -> n # 0 )
78	77	adantl 277	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> n # 0 )
79	66, 69, 62, 78, 63	divcanap7d 8863	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) = ( z / n ) )
80	79	eqeq2d 2208	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) <-> A = ( z / n ) ) )
81	80	biimp3ar 1357	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
82	76, 81	jca 306	. . . . . 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 2874	. . . . 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 6236	. . . . . . 7 |- ( x e. ( ZZ X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) )
85		elxp6 6236	. . . . . . 7 |- ( y e. ( ZZ X. NN ) <-> ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) )
86		simprl 529	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. ZZ )
87	86	ad2antrr 488	. . . . . . . . . . 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 531	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. NN )
89	88	ad2antrr 488	. . . . . . . . . . 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 537	. . . . . . . . . . 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 529	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 1st ` y ) e. ZZ )
92	91	ad2antlr 489	. . . . . . . . . . 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 531	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 2nd ` y ) e. NN )
94	93	ad2antlr 489	. . . . . . . . . . 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 539	. . . . . . . . . . 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 538	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 2231	. . . . . . . . . . 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 12289	. . . . . . . . . . 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 1274	. . . . . . . . . 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 2766	. . . . . . . . . . . 12 |- x e. _V
102		1stexg 6234	. . . . . . . . . . . 12 |- ( x e. _V -> ( 1st ` x ) e. _V )
103	101, 102	ax-mp 5	. . . . . . . . . . 11 |- ( 1st ` x ) e. _V
104		2ndexg 6235	. . . . . . . . . . . 12 |- ( x e. _V -> ( 2nd ` x ) e. _V )
105	101, 104	ax-mp 5	. . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
106	103, 105	opth 4271	. . . . . . . . . 10 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. ( 1st ` y ) , ( 2nd ` y ) >. <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
107	100, 106	sylibr 134	. . . . . . . . 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 533	. . . . . . . . 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 535	. . . . . . . . 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 2239	. . . . . . . 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 115	. . . . . . 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 291	. . . . . 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 2551	. . . . 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 313	. . . 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 1207	. . 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 2620	. 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 9713	. 2 |- ( A e. QQ <-> E. z e. ZZ E. n e. NN A = ( z / n ) )
118		fveq2 5561	. . . . . 6 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
119		fveq2 5561	. . . . . 6 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
120	118, 119	oveq12d 5943	. . . . 5 |- ( x = y -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( 1st ` y ) gcd ( 2nd ` y ) ) )
121	120	eqeq1d 2205	. . . 4 |- ( x = y -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) )
122	118, 119	oveq12d 5943	. . . . 5 |- ( x = y -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
123	122	eqeq2d 2208	. . . 4 |- ( x = y -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) )
124	121, 123	anbi12d 473	. . 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 2958	. 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 201	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   E.wrex 2476   E!wreu 2477   _Vcvv 2763   <.cop 3626   class class class wbr 4034   X. cxp 4662   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   x. cmul 7901   < clt 8078   # cap 8625   / cdiv 8716   NNcn 9007   NN0cn0 9266   ZZcz 9343   QQcq 9710   || cdvds 11969   gcd cgcd 12145

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146

This theorem is referenced by:  qnumdencl  12380  qnumdenbi  12385