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Theorem qredeu 14455
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 10926 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
2 gcddvds 14360 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z  /\  ( z  gcd  n
)  ||  n )
)
32simpld 457 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  z )
41, 3sylan2 472 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  z )
5 gcdcl 14362 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  e.  NN0 )
61, 5sylan2 472 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN0 )
76nn0zd 11005 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  ZZ )
8 simpl 455 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  ZZ )
91adantl 464 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  ZZ )
10 nnne0 10608 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
1110neneqd 2605 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  -.  n  =  0 )
1211intnand 917 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  -.  ( z  =  0  /\  n  =  0 ) )
1312adantl 464 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  -.  ( z  =  0  /\  n  =  0 ) )
14 gcdn0cl 14359 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ZZ  /\  n  e.  ZZ )  /\  -.  ( z  =  0  /\  n  =  0 ) )  ->  ( z  gcd  n )  e.  NN )
158, 9, 13, 14syl21anc 1229 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN )
16 nnne0 10608 . . . . . . . . . . 11  |-  ( ( z  gcd  n )  e.  NN  ->  (
z  gcd  n )  =/=  0 )
1715, 16syl 17 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  =/=  0 )
18 dvdsval2 14196 . . . . . . . . . 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 1230 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  z  <->  ( z  /  ( z  gcd  n ) )  e.  ZZ ) )
204, 19mpbid 210 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  /  (
z  gcd  n )
)  e.  ZZ )
21203adant3 1017 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  ZZ )
222simprd 461 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  n )
231, 22sylan2 472 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  n )
24 dvdsval2 14196 . . . . . . . . . . . 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 1230 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  n  <->  ( n  /  ( z  gcd  n ) )  e.  ZZ ) )
2623, 25mpbid 210 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( n  /  (
z  gcd  n )
)  e.  ZZ )
27 nnre 10582 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  RR )
2827adantl 464 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  RR )
296nn0red 10893 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  RR )
30 nngt0 10604 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  0  <  n )
3130adantl 464 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  n )
32 nngt0 10604 . . . . . . . . . . . 12  |-  ( ( z  gcd  n )  e.  NN  ->  0  <  ( z  gcd  n
) )
3315, 32syl 17 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( z  gcd  n ) )
3428, 29, 31, 33divgt0d 10520 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( n  /  ( z  gcd  n ) ) )
3526, 34jca 530 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
36353adant3 1017 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
37 elnnz 10914 . . . . . . . 8  |-  ( ( n  /  ( z  gcd  n ) )  e.  NN  <->  ( (
n  /  ( z  gcd  n ) )  e.  ZZ  /\  0  <  ( n  /  (
z  gcd  n )
) ) )
3836, 37sylibr 212 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  NN )
39 opelxpi 4854 . . . . . . 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 659 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  <. ( z  /  (
z  gcd  n )
) ,  ( n  /  ( z  gcd  n ) ) >.  e.  ( ZZ  X.  NN ) )
4120, 26gcdcld 14363 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  NN0 )
4241nn0cnd 10894 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  CC )
43 1cnd 9641 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  1  e.  CC )
446nn0cnd 10894 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  CC )
4544mulid1d 9642 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  1 )  =  ( z  gcd  n ) )
46 zcn 10909 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  CC )
4746adantr 463 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  CC )
4847, 44, 17divcan2d 10362 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
z  /  ( z  gcd  n ) ) )  =  z )
49 nncn 10583 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  CC )
5049adantl 464 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  CC )
5150, 44, 17divcan2d 10362 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
n  /  ( z  gcd  n ) ) )  =  n )
5248, 51oveq12d 6295 . . . . . . . . 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 ) )
53 mulgcd 14391 . . . . . . . . . 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
) ) ) ) )
546, 20, 26, 53syl3anc 1230 . . . . . . . . 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 )
) ) ) )
5545, 52, 543eqtr2rd 2450 . . . . . . . 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 ) )
5642, 43, 44, 17, 55mulcanad 10224 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
57563adant3 1017 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
5810adantl 464 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  =/=  0 )
5947, 50, 44, 58, 17divcan7d 10388 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) )  =  ( z  /  n ) )
6059eqeq2d 2416 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) )  <-> 
A  =  ( z  /  n ) ) )
6160biimp3ar 1331 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  A  =  ( (
z  /  ( z  gcd  n ) )  /  ( n  / 
( z  gcd  n
) ) ) )
62 ovex 6305 . . . . . . . . . . 11  |-  ( z  /  ( z  gcd  n ) )  e. 
_V
63 ovex 6305 . . . . . . . . . . 11  |-  ( n  /  ( z  gcd  n ) )  e. 
_V
6462, 63op1std 6793 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 1st `  x
)  =  ( z  /  ( z  gcd  n ) ) )
6562, 63op2ndd 6794 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 2nd `  x
)  =  ( n  /  ( z  gcd  n ) ) )
6664, 65oveq12d 6295 . . . . . . . . 9  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  ( ( z  /  ( z  gcd  n ) )  gcd  ( n  /  (
z  gcd  n )
) ) )
6766eqeq1d 2404 . . . . . . . 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 ) )
6864, 65oveq12d 6295 . . . . . . . . 9  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( ( 1st `  x )  /  ( 2nd `  x ) )  =  ( ( z  /  ( z  gcd  n ) )  / 
( n  /  (
z  gcd  n )
) ) )
6968eqeq2d 2416 . . . . . . . 8  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) )  <->  A  =  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) ) ) )
7067, 69anbi12d 709 . . . . . . 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 ) ) ) ) ) )
7170rspcev 3159 . . . . . 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 ) ) ) )
7240, 57, 61, 71syl12anc 1228 . . . . 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 ) ) ) )
73 elxp6 6815 . . . . . . 7  |-  ( x  e.  ( ZZ  X.  NN )  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) ) )
74 elxp6 6815 . . . . . . 7  |-  ( y  e.  ( ZZ  X.  NN )  <->  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )
75 simprl 756 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 1st `  x )  e.  ZZ )
7675ad2antrr 724 . . . . . . . . . . 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 )
77 simprr 758 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 2nd `  x )  e.  NN )
7877ad2antrr 724 . . . . . . . . . . 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 )
79 simprll 764 . . . . . . . . . . 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 )
80 simprl 756 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 1st `  y )  e.  ZZ )
8180ad2antlr 725 . . . . . . . . . . 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 )
82 simprr 758 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 2nd `  y )  e.  NN )
8382ad2antlr 725 . . . . . . . . . . 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 )
84 simprrl 766 . . . . . . . . . . 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 )
85 simprlr 765 . . . . . . . . . . . 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 ) ) )
86 simprrr 767 . . . . . . . . . . . 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 ) ) )
8785, 86eqtr3d 2445 . . . . . . . . . . 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 ) ) )
88 qredeq 14454 . . . . . . . . . . 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 ) ) )
8976, 78, 79, 81, 83, 84, 87, 88syl331anc 1255 . . . . . . . . . 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
) ) )
90 fvex 5858 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
91 fvex 5858 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
9290, 91opth 4664 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. 
<->  ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  =  ( 2nd `  y
) ) )
9389, 92sylibr 212 . . . . . . . . 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
) >. )
94 simplll 760 . . . . . . . . 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 ) >. )
95 simplrl 762 . . . . . . . . 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 ) >. )
9693, 94, 953eqtr4d 2453 . . . . . . . 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 )
9796ex 432 . . . . . . 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 ) )
9873, 74, 97syl2anb 477 . . . . . 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 ) )
9998rgen2a 2830 . . . . 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 )
10072, 99jctir 536 . . . 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 ) ) )
1011003expia 1199 . . 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 ) ) ) )
102101rexlimivv 2900 . 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 ) ) )
103 elq 11228 . 2  |-  ( A  e.  QQ  <->  E. z  e.  ZZ  E. n  e.  NN  A  =  ( z  /  n ) )
104 fveq2 5848 . . . . . 6  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
105 fveq2 5848 . . . . . 6  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
106104, 105oveq12d 6295 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  gcd  ( 2nd `  x ) )  =  ( ( 1st `  y
)  gcd  ( 2nd `  y ) ) )
107106eqeq1d 2404 . . . 4  |-  ( x  =  y  ->  (
( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  <->  ( ( 1st `  y )  gcd  ( 2nd `  y ) )  =  1 ) )
108104, 105oveq12d 6295 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  /  ( 2nd `  x ) )  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) )
109108eqeq2d 2416 . . . 4  |-  ( x  =  y  ->  ( A  =  ( ( 1st `  x )  / 
( 2nd `  x
) )  <->  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )
110107, 109anbi12d 709 . . 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 ) ) ) ) )
111110reu4 3242 . 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 ) ) )
112102, 103, 1113imtr4i 266 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   E!wreu 2755   <.cop 3977   class class class wbr 4394    X. cxp 4820   ` cfv 5568  (class class class)co 6277   1stc1st 6781   2ndc2nd 6782   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    x. cmul 9526    < clt 9657    / cdiv 10246   NNcn 10575   NN0cn0 10835   ZZcz 10904   QQcq 11226    || cdvds 14193    gcd cgcd 14351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-dvds 14194  df-gcd 14352
This theorem is referenced by:  qnumdencl  14479  qnumdenbi  14484
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