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Theorem qredeu 13099
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.
StepHypRef Expression
1 nnz 10295 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
2 gcddvds 13007 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z  /\  ( z  gcd  n
)  ||  n )
)
32simpld 446 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  z )
41, 3sylan2 461 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  z )
5 gcdcl 13009 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  e.  NN0 )
61, 5sylan2 461 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN0 )
76nn0zd 10365 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  ZZ )
8 simpl 444 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  ZZ )
91adantl 453 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  ZZ )
10 nnne0 10024 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
1110neneqd 2614 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  -.  n  =  0 )
1211intnand 883 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  -.  ( z  =  0  /\  n  =  0 ) )
1312adantl 453 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  -.  ( z  =  0  /\  n  =  0 ) )
14 gcdn0cl 13006 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ZZ  /\  n  e.  ZZ )  /\  -.  ( z  =  0  /\  n  =  0 ) )  ->  ( z  gcd  n )  e.  NN )
158, 9, 13, 14syl21anc 1183 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN )
16 nnne0 10024 . . . . . . . . . . 11  |-  ( ( z  gcd  n )  e.  NN  ->  (
z  gcd  n )  =/=  0 )
1715, 16syl 16 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  =/=  0 )
18 dvdsval2 12847 . . . . . . . . . 10  |-  ( ( ( z  gcd  n
)  e.  ZZ  /\  ( z  gcd  n
)  =/=  0  /\  z  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z 
<->  ( z  /  (
z  gcd  n )
)  e.  ZZ ) )
197, 17, 8, 18syl3anc 1184 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  z  <->  ( z  /  ( z  gcd  n ) )  e.  ZZ ) )
204, 19mpbid 202 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  /  (
z  gcd  n )
)  e.  ZZ )
21203adant3 977 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  ZZ )
222simprd 450 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  n )
231, 22sylan2 461 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  n )
24 dvdsval2 12847 . . . . . . . . . . . 12  |-  ( ( ( z  gcd  n
)  e.  ZZ  /\  ( z  gcd  n
)  =/=  0  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  n 
<->  ( n  /  (
z  gcd  n )
)  e.  ZZ ) )
257, 17, 9, 24syl3anc 1184 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  n  <->  ( n  /  ( z  gcd  n ) )  e.  ZZ ) )
2623, 25mpbid 202 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( n  /  (
z  gcd  n )
)  e.  ZZ )
27 nnre 9999 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  RR )
2827adantl 453 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  RR )
296nn0red 10267 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  RR )
30 nngt0 10021 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  0  <  n )
3130adantl 453 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  n )
32 nngt0 10021 . . . . . . . . . . . 12  |-  ( ( z  gcd  n )  e.  NN  ->  0  <  ( z  gcd  n
) )
3315, 32syl 16 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( z  gcd  n ) )
3428, 29, 31, 33divgt0d 9938 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( n  /  ( z  gcd  n ) ) )
3526, 34jca 519 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
36353adant3 977 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
37 elnnz 10284 . . . . . . . 8  |-  ( ( n  /  ( z  gcd  n ) )  e.  NN  <->  ( (
n  /  ( z  gcd  n ) )  e.  ZZ  /\  0  <  ( n  /  (
z  gcd  n )
) ) )
3836, 37sylibr 204 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  NN )
39 opelxpi 4902 . . . . . . 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 ) )
4021, 38, 39syl2anc 643 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  <. ( z  /  (
z  gcd  n )
) ,  ( n  /  ( z  gcd  n ) ) >.  e.  ( ZZ  X.  NN ) )
4120, 26gcdcld 13010 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  NN0 )
4241nn0cnd 10268 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  CC )
43 ax-1cn 9040 . . . . . . . . 9  |-  1  e.  CC
4443a1i 11 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  1  e.  CC )
456nn0cnd 10268 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  CC )
4645mulid1d 9097 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  1 )  =  ( z  gcd  n ) )
47 zcn 10279 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  CC )
4847adantr 452 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  CC )
4948, 45, 17divcan2d 9784 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
z  /  ( z  gcd  n ) ) )  =  z )
50 nncn 10000 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  CC )
5150adantl 453 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  CC )
5251, 45, 17divcan2d 9784 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
n  /  ( z  gcd  n ) ) )  =  n )
5349, 52oveq12d 6091 . . . . . . . . 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 ) )
54 mulgcd 13038 . . . . . . . . . 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
) ) ) ) )
556, 20, 26, 54syl3anc 1184 . . . . . . . . 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 )
) ) ) )
5646, 53, 553eqtr2rd 2474 . . . . . . . 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 ) )
5742, 44, 45, 17, 56mulcanad 9649 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
58573adant3 977 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
5910adantl 453 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  =/=  0 )
6048, 51, 45, 59, 17divcan7d 9810 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) )  =  ( z  /  n ) )
6160eqeq2d 2446 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) )  <-> 
A  =  ( z  /  n ) ) )
6261biimp3ar 1284 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  A  =  ( (
z  /  ( z  gcd  n ) )  /  ( n  / 
( z  gcd  n
) ) ) )
63 ovex 6098 . . . . . . . . . . 11  |-  ( z  /  ( z  gcd  n ) )  e. 
_V
64 ovex 6098 . . . . . . . . . . 11  |-  ( n  /  ( z  gcd  n ) )  e. 
_V
6563, 64op1std 6349 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 1st `  x
)  =  ( z  /  ( z  gcd  n ) ) )
6663, 64op2ndd 6350 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 2nd `  x
)  =  ( n  /  ( z  gcd  n ) ) )
6765, 66oveq12d 6091 . . . . . . . . 9  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  ( ( z  /  ( z  gcd  n ) )  gcd  ( n  /  (
z  gcd  n )
) ) )
6867eqeq1d 2443 . . . . . . . 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 ) )
6965, 66oveq12d 6091 . . . . . . . . 9  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( ( 1st `  x )  /  ( 2nd `  x ) )  =  ( ( z  /  ( z  gcd  n ) )  / 
( n  /  (
z  gcd  n )
) ) )
7069eqeq2d 2446 . . . . . . . 8  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) )  <->  A  =  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) ) ) )
7168, 70anbi12d 692 . . . . . . 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 ) ) ) ) ) )
7271rspcev 3044 . . . . . 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 ) ) ) )
7340, 58, 62, 72syl12anc 1182 . . . . 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 ) ) ) )
74 elxp6 6370 . . . . . . 7  |-  ( x  e.  ( ZZ  X.  NN )  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) ) )
75 elxp6 6370 . . . . . . 7  |-  ( y  e.  ( ZZ  X.  NN )  <->  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )
76 simprl 733 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 1st `  x )  e.  ZZ )
7776ad2antrr 707 . . . . . . . . . . 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 )
78 simprr 734 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 2nd `  x )  e.  NN )
7978ad2antrr 707 . . . . . . . . . . 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 )
80 simprll 739 . . . . . . . . . . 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 )
81 simprl 733 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 1st `  y )  e.  ZZ )
8281ad2antlr 708 . . . . . . . . . . 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 )
83 simprr 734 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 2nd `  y )  e.  NN )
8483ad2antlr 708 . . . . . . . . . . 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 )
85 simprrl 741 . . . . . . . . . . 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 )
86 simprlr 740 . . . . . . . . . . . 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 ) ) )
87 simprrr 742 . . . . . . . . . . . 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 ) ) )
8886, 87eqtr3d 2469 . . . . . . . . . . 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 ) ) )
89 qredeq 13098 . . . . . . . . . . 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 ) ) )
9077, 79, 80, 82, 84, 85, 88, 89syl331anc 1209 . . . . . . . . . 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
) ) )
91 fvex 5734 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
92 fvex 5734 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
9391, 92opth 4427 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. 
<->  ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  =  ( 2nd `  y
) ) )
9490, 93sylibr 204 . . . . . . . . 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
) >. )
95 simplll 735 . . . . . . . . 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 ) >. )
96 simplrl 737 . . . . . . . . 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 ) >. )
9794, 95, 963eqtr4d 2477 . . . . . . . 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 )
9897ex 424 . . . . . . 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 ) )
9974, 75, 98syl2anb 466 . . . . . 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 ) )
10099rgen2a 2764 . . . . 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 )
10173, 100jctir 525 . . . 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 ) ) )
1021013expia 1155 . . 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 ) ) ) )
103102rexlimivv 2827 . 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 ) ) )
104 elq 10568 . 2  |-  ( A  e.  QQ  <->  E. z  e.  ZZ  E. n  e.  NN  A  =  ( z  /  n ) )
105 fveq2 5720 . . . . . 6  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
106 fveq2 5720 . . . . . 6  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
107105, 106oveq12d 6091 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  gcd  ( 2nd `  x ) )  =  ( ( 1st `  y
)  gcd  ( 2nd `  y ) ) )
108107eqeq1d 2443 . . . 4  |-  ( x  =  y  ->  (
( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  <->  ( ( 1st `  y )  gcd  ( 2nd `  y ) )  =  1 ) )
109105, 106oveq12d 6091 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  /  ( 2nd `  x ) )  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) )
110109eqeq2d 2446 . . . 4  |-  ( x  =  y  ->  ( A  =  ( ( 1st `  x )  / 
( 2nd `  x
) )  <->  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )
111108, 110anbi12d 692 . . 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 ) ) ) ) )
112111reu4 3120 . 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 ) ) )
113103, 104, 1123imtr4i 258 1  |-  ( A  e.  QQ  ->  E! x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699   <.cop 3809   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   QQcq 10566    || cdivides 12844    gcd cgcd 12998
This theorem is referenced by:  qnumdencl  13123  qnumdenbi  13128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999
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