Theorem qredeu 15372

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 11399	. . . . . . . . . 10 |- ( n e. NN -> n e. ZZ )
2		gcddvds 15225	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( ( z gcd n ) || z /\ ( z gcd n ) || n ) )
3	2	simpld 475	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || z )
4	1, 3	sylan2 491	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || z )
5		gcdcl 15228	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) e. NN0 )
6	1, 5	sylan2 491	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN0 )
7	6	nn0zd 11480	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. ZZ )
8		simpl 473	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> z e. ZZ )
9	1	adantl 482	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> n e. ZZ )
10		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( n e. NN -> n =/= 0 )
11	10	neneqd 2799	. . . . . . . . . . . . . 14 |- ( n e. NN -> -. n = 0 )
12	11	intnand 962	. . . . . . . . . . . . 13 |- ( n e. NN -> -. ( z = 0 /\ n = 0 ) )
13	12	adantl 482	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. NN ) -> -. ( z = 0 /\ n = 0 ) )
14		gcdn0cl 15224	. . . . . . . . . . . 12 |- ( ( ( z e. ZZ /\ n e. ZZ ) /\ -. ( z = 0 /\ n = 0 ) ) -> ( z gcd n ) e. NN )
15	8, 9, 13, 14	syl21anc 1325	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. NN )
16		nnne0 11053	. . . . . . . . . . 11 |- ( ( z gcd n ) e. NN -> ( z gcd n ) =/= 0 )
17	15, 16	syl 17	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) =/= 0 )
18		dvdsval2 14986	. . . . . . . . . 10 |- ( ( ( z gcd n ) e. ZZ /\ ( z gcd n ) =/= 0 /\ z e. ZZ ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
19	7, 17, 8, 18	syl3anc 1326	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || z <-> ( z / ( z gcd n ) ) e. ZZ ) )
20	4, 19	mpbid 222	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( z / ( z gcd n ) ) e. ZZ )
21	20	3adant3 1081	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( z / ( z gcd n ) ) e. ZZ )
22	2	simprd 479	. . . . . . . . . . . 12 |- ( ( z e. ZZ /\ n e. ZZ ) -> ( z gcd n ) || n )
23	1, 22	sylan2 491	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) || n )
24		dvdsval2 14986	. . . . . . . . . . . 12 |- ( ( ( z gcd n ) e. ZZ /\ ( z gcd n ) =/= 0 /\ n e. ZZ ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
25	7, 17, 9, 24	syl3anc 1326	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) || n <-> ( n / ( z gcd n ) ) e. ZZ ) )
26	23, 25	mpbid 222	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( n / ( z gcd n ) ) e. ZZ )
27		nnre 11027	. . . . . . . . . . . 12 |- ( n e. NN -> n e. RR )
28	27	adantl 482	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. RR )
29	6	nn0red 11352	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. RR )
30		nngt0 11049	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
31	30	adantl 482	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < n )
32		nngt0 11049	. . . . . . . . . . . 12 |- ( ( z gcd n ) e. NN -> 0 < ( z gcd n ) )
33	15, 32	syl 17	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( z gcd n ) )
34	28, 29, 31, 33	divgt0d 10959	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> 0 < ( n / ( z gcd n ) ) )
35	26, 34	jca 554	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
36	35	3adant3 1081	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
37		elnnz 11387	. . . . . . . 8 |- ( ( n / ( z gcd n ) ) e. NN <-> ( ( n / ( z gcd n ) ) e. ZZ /\ 0 < ( n / ( z gcd n ) ) ) )
38	36, 37	sylibr 224	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( n / ( z gcd n ) ) e. NN )
39	21, 38	opelxpd 5149	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) )
40	20, 26	gcdcld 15230	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. NN0 )
41	40	nn0cnd 11353	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) e. CC )
42		1cnd 10056	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> 1 e. CC )
43	6	nn0cnd 11353	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( z gcd n ) e. CC )
44	43	mulid1d 10057	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. 1 ) = ( z gcd n ) )
45		zcn 11382	. . . . . . . . . . . 12 |- ( z e. ZZ -> z e. CC )
46	45	adantr 481	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> z e. CC )
47	46, 43, 17	divcan2d 10803	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( z / ( z gcd n ) ) ) = z )
48		nncn 11028	. . . . . . . . . . . 12 |- ( n e. NN -> n e. CC )
49	48	adantl 482	. . . . . . . . . . 11 |- ( ( z e. ZZ /\ n e. NN ) -> n e. CC )
50	49, 43, 17	divcan2d 10803	. . . . . . . . . 10 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( n / ( z gcd n ) ) ) = n )
51	47, 50	oveq12d 6668	. . . . . . . . 9 |- ( ( 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 ) )
52		mulgcd 15265	. . . . . . . . . 10 |- ( ( ( 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 ) ) ) ) )
53	6, 20, 26, 52	syl3anc 1326	. . . . . . . . 9 |- ( ( 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 ) ) ) ) )
54	44, 51, 53	3eqtr2rd 2663	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z gcd n ) x. ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) ) = ( ( z gcd n ) x. 1 ) )
55	41, 42, 43, 17, 54	mulcanad 10662	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
56	55	3adant3 1081	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 )
57	10	adantl 482	. . . . . . . . 9 |- ( ( z e. ZZ /\ n e. NN ) -> n =/= 0 )
58	46, 49, 43, 57, 17	divcan7d 10829	. . . . . . . 8 |- ( ( z e. ZZ /\ n e. NN ) -> ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) = ( z / n ) )
59	58	eqeq2d 2632	. . . . . . 7 |- ( ( z e. ZZ /\ n e. NN ) -> ( A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) <-> A = ( z / n ) ) )
60	59	biimp3ar 1433	. . . . . 6 |- ( ( z e. ZZ /\ n e. NN /\ A = ( z / n ) ) -> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
61		ovex 6678	. . . . . . . . . . 11 |- ( z / ( z gcd n ) ) e. _V
62		ovex 6678	. . . . . . . . . . 11 |- ( n / ( z gcd n ) ) e. _V
63	61, 62	op1std 7178	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 1st ` x ) = ( z / ( z gcd n ) ) )
64	61, 62	op2ndd 7179	. . . . . . . . . 10 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( 2nd ` x ) = ( n / ( z gcd n ) ) )
65	63, 64	oveq12d 6668	. . . . . . . . 9 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) )
66	65	eqeq1d 2624	. . . . . . . 8 |- ( 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 ) )
67	63, 64	oveq12d 6668	. . . . . . . . 9 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) )
68	67	eqeq2d 2632	. . . . . . . 8 |- ( x = <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) ) )
69	66, 68	anbi12d 747	. . . . . . 7 |- ( 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 ) ) ) ) ) )
70	69	rspcev 3309	. . . . . 6 |- ( ( <. ( z / ( z gcd n ) ) , ( n / ( z gcd n ) ) >. e. ( ZZ X. NN ) /\ ( ( ( z / ( z gcd n ) ) gcd ( n / ( z gcd n ) ) ) = 1 /\ A = ( ( z / ( z gcd n ) ) / ( n / ( z gcd n ) ) ) ) ) -> E. x e. ( ZZ X. NN ) ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 /\ A = ( ( 1st ` x ) / ( 2nd ` x ) ) ) )
71	39, 56, 60, 70	syl12anc 1324	. . . . 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 ) ) ) )
72		elxp6 7200	. . . . . . 7 |- ( x e. ( ZZ X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) )
73		elxp6 7200	. . . . . . 7 |- ( y e. ( ZZ X. NN ) <-> ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) )
74		simprl 794	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. ZZ )
75	74	ad2antrr 762	. . . . . . . . . . 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 )
76		simprr 796	. . . . . . . . . . . 12 |- ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ZZ /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. NN )
77	76	ad2antrr 762	. . . . . . . . . . 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 )
78		simprll 802	. . . . . . . . . . 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 )
79		simprl 794	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 1st ` y ) e. ZZ )
80	79	ad2antlr 763	. . . . . . . . . . 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 )
81		simprr 796	. . . . . . . . . . . 12 |- ( ( y = <. ( 1st ` y ) , ( 2nd ` y ) >. /\ ( ( 1st ` y ) e. ZZ /\ ( 2nd ` y ) e. NN ) ) -> ( 2nd ` y ) e. NN )
82	81	ad2antlr 763	. . . . . . . . . . 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 )
83		simprrl 804	. . . . . . . . . . 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 )
84		simprlr 803	. . . . . . . . . . . 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 ) ) )
85		simprrr 805	. . . . . . . . . . . 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 ) ) )
86	84, 85	eqtr3d 2658	. . . . . . . . . . 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 ) ) )
87		qredeq 15371	. . . . . . . . . . 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 ) ) )
88	75, 77, 78, 80, 82, 83, 86, 87	syl331anc 1351	. . . . . . . . . 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 ) ) )
89		fvex 6201	. . . . . . . . . . 11 |- ( 1st ` x ) e. _V
90		fvex 6201	. . . . . . . . . . 11 |- ( 2nd ` x ) e. _V
91	89, 90	opth 4945	. . . . . . . . . 10 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. ( 1st ` y ) , ( 2nd ` y ) >. <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) )
92	88, 91	sylibr 224	. . . . . . . . 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 ) >. )
93		simplll 798	. . . . . . . . 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 ) >. )
94		simplrl 800	. . . . . . . . 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 ) >. )
95	92, 93, 94	3eqtr4d 2666	. . . . . . . 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 )
96	95	ex 450	. . . . . . 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 ) )
97	72, 73, 96	syl2anb 496	. . . . . 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 ) )
98	97	rgen2a 2977	. . . . 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 )
99	71, 98	jctir 561	. . . 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 ) ) )
100	99	3expia 1267	. . 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 ) ) ) )
101	100	rexlimivv 3036	. 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 ) ) )
102		elq 11790	. 2 |- ( A e. QQ <-> E. z e. ZZ E. n e. NN A = ( z / n ) )
103		fveq2 6191	. . . . . 6 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
104		fveq2 6191	. . . . . 6 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
105	103, 104	oveq12d 6668	. . . . 5 |- ( x = y -> ( ( 1st ` x ) gcd ( 2nd ` x ) ) = ( ( 1st ` y ) gcd ( 2nd ` y ) ) )
106	105	eqeq1d 2624	. . . 4 |- ( x = y -> ( ( ( 1st ` x ) gcd ( 2nd ` x ) ) = 1 <-> ( ( 1st ` y ) gcd ( 2nd ` y ) ) = 1 ) )
107	103, 104	oveq12d 6668	. . . . 5 |- ( x = y -> ( ( 1st ` x ) / ( 2nd ` x ) ) = ( ( 1st ` y ) / ( 2nd ` y ) ) )
108	107	eqeq2d 2632	. . . 4 |- ( x = y -> ( A = ( ( 1st ` x ) / ( 2nd ` x ) ) <-> A = ( ( 1st ` y ) / ( 2nd ` y ) ) ) )
109	106, 108	anbi12d 747	. . 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 ) ) ) ) )
110	109	reu4 3400	. 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 ) ) )
111	101, 102, 110	3imtr4i 281	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   <.cop 4183   class class class wbr 4653   X. cxp 5112   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   QQcq 11788   || cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  qnumdencl  15447  qnumdenbi  15452