Theorem qredeu 11778

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 9073	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
2		gcddvds 11652	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( ( z gcd n ) || z /\ ( z gcd n ) || n ) )
3	2	simpld 111	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || z )
4	1, 3	sylan2 284	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || z )
5		gcdcl 11655	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) e. NN0 )
6	1, 5	sylan2 284	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN0 )
7	6	nn0zd 9171	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. ZZ )
8		simpl 108	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. ZZ )
9	1	adantl 275	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. ZZ )
10		nnne0 8748	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
11	10	neneqd 2329	. . . . . . . . . . . . . 14 |- ( n e. NN -> -. n = 0 )
12	11	intnand 916	. . . . . . . . . . . . 13 |- ( n e. NN -> -. ( z = 0 /\ n = 0 ) )
13	12	adantl 275	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> -. ( z = 0 /\ n = 0 ) )
14		gcdn0cl 11651	. . . . . . . . . . . 12 |- ( ( ( z e. ZZ /\ n e. ZZ ) /\ -. ( z = 0 /\ n = 0 ) ) -> ( z gcd n ) e. NN )
15	8, 9, 13, 14	syl21anc 1215	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN )
16	15	nnne0d 8765	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) =/= 0 )
17		dvdsval2 11496	. . . . . . . . . 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 1216	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
19	4, 18	mpbid 146	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( z / ( z gcd n ) ) e. ZZ )
20	19	3adant3 1001	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. ZZ )
21	2	simprd 113	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || n )
22	1, 21	sylan2 284	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || n )
23		dvdsval2 11496	. . . . . . . . . . . 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 1216	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
25	22, 24	mpbid 146	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( n / ( z gcd n ) ) e. ZZ )
26		nnre 8727	. . . . . . . . . . . 12 |- ( n e. NN -> n e. RR )
27	26	adantl 275	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. RR )
28	6	nn0red 9031	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. RR )
29		nngt0 8745	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
30	29	adantl 275	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < n )
31	15	nngt0d 8764	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( z gcd n ) )
32	27, 28, 30, 31	divgt0d 8693	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( n / ( z gcd n ) ) )
33	25, 32	jca 304	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
34	33	3adant3 1001	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
35		elnnz 9064	. . . . . . . 8 |- ( ( n / ( z gcd n ) ) e. NN <-> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
36	34, 35	sylibr 133	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. NN )
37		opelxpi 4571	. . . . . . 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 408	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) )
39		fveq2 5421	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 1st ` x ) = ( 1st ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
40		simp1 981	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> z e. ZZ )
41	15	3adant3 1001	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z gcd n ) e. NN )
42		znq 9416	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ ( z gcd n ) e. NN ) -> ( z / ( z gcd n ) ) e. QQ )
43	40, 41, 42	syl2anc 408	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. QQ )
44	9	3adant3 1001	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> n e. ZZ )
45		znq 9416	. . . . . . . . . . . 12 |- ( ( n e. ZZ /\ ( z gcd n ) e. NN ) -> ( n / ( z gcd n ) ) e. QQ )
46	44, 41, 45	syl2anc 408	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. QQ )
47		op1stg 6048	. . . . . . . . . . 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 408	. . . . . . . . . 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 2194	. . . . . . . . 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 5421	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 2nd ` x ) = ( 2nd ` <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. ) )
51		op2ndg 6049	. . . . . . . . . . 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 408	. . . . . . . . . 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 2194	. . . . . . . . 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 5792	. . . . . . . 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 2148	. . . . . . 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 5792	. . . . . . . 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 2151	. . . . . . 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 464	. . . . . 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 11657	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. NN0 )
60	59	nn0cnd 9032	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. CC )
61		1cnd 7782	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> 1 e. CC )
62	6	nn0cnd 9032	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. CC )
63	15	nnap0d 8766	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) # 0 )
64	62	mulid1d 7783	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. 1 ) = ( z gcd n ) )
65		zcn 9059	. . . . . . . . . . . . 13 |- ( z e. ZZ -> z e. CC )
66	65	adantr 274	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. CC )
67	66, 62, 63	divcanap2d 8552	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) = z )
68		nncn 8728	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. CC )
69	68	adantl 275	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. CC )
70	69, 62, 63	divcanap2d 8552	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) = n )
71	67, 70	oveq12d 5792	. . . . . . . . . 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 11704	. . . . . . . . . . 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 1216	. . . . . . . . . 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 2179	. . . . . . . . 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 8424	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
76	75	3adant3 1001	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
77		nnap0 8749	. . . . . . . . . . 11 |- ( n e. NN -> n # 0 )
78	77	adantl 275	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> n # 0 )
79	66, 69, 62, 78, 63	divcanap7d 8579	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) = ( z / n ) )
80	79	eqeq2d 2151	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) <-> A = ( z / n ) ) )
81	80	biimp3ar 1324	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
82	76, 81	jca 304	. . . . . 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 2795	. . . . 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 6067	. . . . . . 7 |- ( x e. ( ZZ X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) )
85		elxp6 6067	. . . . . . 7 |- ( y e. ( ZZ X. NN ) <-> ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) )
86		simprl 520	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. ZZ )
87	86	ad2antrr 479	. . . . . . . . . . 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 521	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. NN )
89	88	ad2antrr 479	. . . . . . . . . . 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 526	. . . . . . . . . . 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 520	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 1st ` y ) e. ZZ )
92	91	ad2antlr 480	. . . . . . . . . . 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 521	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 2nd ` y ) e. NN )
94	93	ad2antlr 480	. . . . . . . . . . 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 528	. . . . . . . . . . 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 527	. . . . . . . . . . . 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 529	. . . . . . . . . . . 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 2174	. . . . . . . . . . 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 11777	. . . . . . . . . . 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 1241	. . . . . . . . . 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 2689	. . . . . . . . . . . 12 |- x e. _V
102		1stexg 6065	. . . . . . . . . . . 12 |- ( x e. _V -> ( 1st ` x ) e. _V )
103	101, 102	ax-mp 5	. . . . . . . . . . 11 |- ( 1st ` x ) e. _V
104		2ndexg 6066	. . . . . . . . . . . 12 |- ( x e. _V -> ( 2nd ` x ) e. _V )
105	101, 104	ax-mp 5	. . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
106	103, 105	opth 4159	. . . . . . . . . 10 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. ( 1st ` y ) , ( 2nd ` y ) >. <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
107	100, 106	sylibr 133	. . . . . . . . 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 522	. . . . . . . . 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 524	. . . . . . . . 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 2182	. . . . . . . 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 114	. . . . . . 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 289	. . . . . 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 2486	. . . . 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 311	. . . 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 1183	. . 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 2555	. 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 9414	. 2 |- ( A e. QQ <-> E. z e. ZZ E. n e. NN A = ( z / n ) )
118		fveq2 5421	. . . . . 6 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
119		fveq2 5421	. . . . . 6 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
120	118, 119	oveq12d 5792	. . . . 5 |- ( x = y -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( 1st ` y ) gcd ( 2nd ` y ) ) )
121	120	eqeq1d 2148	. . . 4 |- ( x = y -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) )
122	118, 119	oveq12d 5792	. . . . 5 |- ( x = y -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
123	122	eqeq2d 2151	. . . 4 |- ( x = y -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) )
124	121, 123	anbi12d 464	. . 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 2878	. 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 200	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 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   E.wrex 2417   E!wreu 2418   _Vcvv 2686   <.cop 3530   class class class wbr 3929   X. cxp 4537   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   x. cmul 7625   < clt 7800   # cap 8343   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZcz 9054   QQcq 9411   || cdvds 11493   gcd cgcd 11635

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494  df-gcd 11636

This theorem is referenced by:  qnumdencl  11865  qnumdenbi  11870