Theorem qredeu 12132

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 9303	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
2		gcddvds 11999	. . . . . . . . . . 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 12002	. . . . . . . . . . . 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 9404	. . . . . . . . . 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 8978	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
11	10	neneqd 2381	. . . . . . . . . . . . . 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 11998	. . . . . . . . . . . 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 8995	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) =/= 0 )
17		dvdsval2 11832	. . . . . . . . . 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 11832	. . . . . . . . . . . 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 8957	. . . . . . . . . . . 12 |- ( n e. NN -> n e. RR )
27	26	adantl 277	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. RR )
28	6	nn0red 9261	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. RR )
29		nngt0 8975	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
30	29	adantl 277	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < n )
31	15	nngt0d 8994	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( z gcd n ) )
32	27, 28, 30, 31	divgt0d 8923	. . . . . . . . . 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 9294	. . . . . . . 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 4676	. . . . . . 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 5534	. . . . . . . . . 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 9656	. . . . . . . . . . . 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 9656	. . . . . . . . . . . 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 6176	. . . . . . . . . . 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 2244	. . . . . . . . 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 5534	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 2nd ` x ) = ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
51		op2ndg 6177	. . . . . . . . . . 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 2244	. . . . . . . . 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 5915	. . . . . . . 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 2198	. . . . . . 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 5915	. . . . . . . 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 2201	. . . . . . 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 12004	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. NN0 )
60	59	nn0cnd 9262	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. CC )
61		1cnd 8004	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> 1 e. CC )
62	6	nn0cnd 9262	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. CC )
63	15	nnap0d 8996	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) # 0 )
64	62	mulridd 8005	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. 1 ) = ( z gcd n ) )
65		zcn 9289	. . . . . . . . . . . . 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 8780	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) = z )
68		nncn 8958	. . . . . . . . . . . . 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 8780	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) = n )
71	67, 70	oveq12d 5915	. . . . . . . . . 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 12052	. . . . . . . . . . 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 2229	. . . . . . . . 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 8651	. . . . . . . 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 8979	. . . . . . . . . . 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 8807	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) = ( z / n ) )
80	79	eqeq2d 2201	. . . . . . . 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 2862	. . . . 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 6195	. . . . . . 7 |- ( x e. ( ZZ X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) )
85		elxp6 6195	. . . . . . 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 2224	. . . . . . . . . . 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 12131	. . . . . . . . . . 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 2755	. . . . . . . . . . . 12 |- x e. _V
102		1stexg 6193	. . . . . . . . . . . 12 |- ( x e. _V -> ( 1st ` x ) e. _V )
103	101, 102	ax-mp 5	. . . . . . . . . . 11 |- ( 1st ` x ) e. _V
104		2ndexg 6194	. . . . . . . . . . . 12 |- ( x e. _V -> ( 2nd ` x ) e. _V )
105	101, 104	ax-mp 5	. . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
106	103, 105	opth 4255	. . . . . . . . . 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 2232	. . . . . . . 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 2544	. . . . 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 2613	. 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 9654	. 2 |- ( A e. QQ <-> E. z e. ZZ E. n e. NN A = ( z / n ) )
118		fveq2 5534	. . . . . 6 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
119		fveq2 5534	. . . . . 6 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
120	118, 119	oveq12d 5915	. . . . 5 |- ( x = y -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( 1st ` y ) gcd ( 2nd ` y ) ) )
121	120	eqeq1d 2198	. . . 4 |- ( x = y -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) )
122	118, 119	oveq12d 5915	. . . . 5 |- ( x = y -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
123	122	eqeq2d 2201	. . . 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 2946	. 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 2160   =/= wne 2360   A.wral 2468   E.wrex 2469   E!wreu 2470   _Vcvv 2752   <.cop 3610   class class class wbr 4018   X. cxp 4642   ` cfv 5235  (class class class)co 5897   1stc1st 6164   2ndc2nd 6165   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843   x. cmul 7847   < clt 8023   # cap 8569   / cdiv 8660   NNcn 8950   NN0cn0 9207   ZZcz 9284   QQcq 9651   || cdvds 11829   gcd cgcd 11978

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-sup 7014  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-fl 10303  df-mod 10356  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-dvds 11830  df-gcd 11979

This theorem is referenced by:  qnumdencl  12222  qnumdenbi  12227