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Theorem qredeu 11571
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 8925 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
2 gcddvds 11447 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z  /\  ( z  gcd  n
)  ||  n )
)
32simpld 111 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  z )
41, 3sylan2 282 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  z )
5 gcdcl 11450 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  e.  NN0 )
61, 5sylan2 282 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN0 )
76nn0zd 9023 . . . . . . . . . 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 )
91adantl 273 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  ZZ )
10 nnne0 8606 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
1110neneqd 2288 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  -.  n  =  0 )
1211intnand 884 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  -.  ( z  =  0  /\  n  =  0 ) )
1312adantl 273 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  -.  ( z  =  0  /\  n  =  0 ) )
14 gcdn0cl 11446 . . . . . . . . . . . 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 )
1615nnne0d 8623 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  =/=  0 )
17 dvdsval2 11291 . . . . . . . . . 10  |-  ( ( ( z  gcd  n
)  e.  ZZ  /\  ( z  gcd  n
)  =/=  0  /\  z  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z 
<->  ( z  /  (
z  gcd  n )
)  e.  ZZ ) )
187, 16, 8, 17syl3anc 1184 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  z  <->  ( z  /  ( z  gcd  n ) )  e.  ZZ ) )
194, 18mpbid 146 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  /  (
z  gcd  n )
)  e.  ZZ )
20193adant3 969 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  ZZ )
212simprd 113 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  n )
221, 21sylan2 282 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  n )
23 dvdsval2 11291 . . . . . . . . . . . 12  |-  ( ( ( z  gcd  n
)  e.  ZZ  /\  ( z  gcd  n
)  =/=  0  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  n 
<->  ( n  /  (
z  gcd  n )
)  e.  ZZ ) )
247, 16, 9, 23syl3anc 1184 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  n  <->  ( n  /  ( z  gcd  n ) )  e.  ZZ ) )
2522, 24mpbid 146 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( n  /  (
z  gcd  n )
)  e.  ZZ )
26 nnre 8585 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  RR )
2726adantl 273 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  RR )
286nn0red 8883 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  RR )
29 nngt0 8603 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  0  <  n )
3029adantl 273 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  n )
3115nngt0d 8622 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( z  gcd  n ) )
3227, 28, 30, 31divgt0d 8551 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( n  /  ( z  gcd  n ) ) )
3325, 32jca 302 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
34333adant3 969 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
35 elnnz 8916 . . . . . . . 8  |-  ( ( n  /  ( z  gcd  n ) )  e.  NN  <->  ( (
n  /  ( z  gcd  n ) )  e.  ZZ  /\  0  <  ( n  /  (
z  gcd  n )
) ) )
3634, 35sylibr 133 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  NN )
37 opelxpi 4509 . . . . . . 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 ) )
3820, 36, 37syl2anc 406 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  <. ( z  /  (
z  gcd  n )
) ,  ( n  /  ( z  gcd  n ) ) >.  e.  ( ZZ  X.  NN ) )
39 fveq2 5353 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 1st `  x
)  =  ( 1st `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
40 simp1 949 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
z  e.  ZZ )
41153adant3 969 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  gcd  n
)  e.  NN )
42 znq 9266 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( z  / 
( z  gcd  n
) )  e.  QQ )
4340, 41, 42syl2anc 406 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  QQ )
4493adant3 969 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  n  e.  ZZ )
45 znq 9266 . . . . . . . . . . . 12  |-  ( ( n  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( n  / 
( z  gcd  n
) )  e.  QQ )
4644, 41, 45syl2anc 406 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  QQ )
47 op1stg 5979 . . . . . . . . . . 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
) ) )
4843, 46, 47syl2anc 406 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( 1st `  <. ( z  /  ( z  gcd  n ) ) ,  ( n  / 
( z  gcd  n
) ) >. )  =  ( z  / 
( z  gcd  n
) ) )
4939, 48sylan9eqr 2154 . . . . . . . . 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 5353 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
51 op2ndg 5980 . . . . . . . . . . 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
) ) )
5243, 46, 51syl2anc 406 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( 2nd `  <. ( z  /  ( z  gcd  n ) ) ,  ( n  / 
( z  gcd  n
) ) >. )  =  ( n  / 
( z  gcd  n
) ) )
5350, 52sylan9eqr 2154 . . . . . . . . 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 )
) )
5449, 53oveq12d 5724 . . . . . . . 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 ) ) ) )
5554eqeq1d 2108 . . . . . . 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 ) )
5649, 53oveq12d 5724 . . . . . . . 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 ) ) ) )
5756eqeq2d 2111 . . . . . . 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 ) ) ) ) )
5855, 57anbi12d 460 . . . . . 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 ) ) ) ) ) )
5919, 25gcdcld 11452 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  NN0 )
6059nn0cnd 8884 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  CC )
61 1cnd 7654 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  1  e.  CC )
626nn0cnd 8884 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  CC )
6315nnap0d 8624 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
) #  0 )
6462mulid1d 7655 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  1 )  =  ( z  gcd  n ) )
65 zcn 8911 . . . . . . . . . . . . 13  |-  ( z  e.  ZZ  ->  z  e.  CC )
6665adantr 272 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  CC )
6766, 62, 63divcanap2d 8413 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
z  /  ( z  gcd  n ) ) )  =  z )
68 nncn 8586 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  CC )
6968adantl 273 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  CC )
7069, 62, 63divcanap2d 8413 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
n  /  ( z  gcd  n ) ) )  =  n )
7167, 70oveq12d 5724 . . . . . . . . . 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 11497 . . . . . . . . . . 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
) ) ) ) )
736, 19, 25, 72syl3anc 1184 . . . . . . . . . 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 )
) ) ) )
7464, 71, 733eqtr2rd 2139 . . . . . . . . 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 ) )
7560, 61, 62, 63, 74mulcanapad 8285 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
76753adant3 969 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
77 nnap0 8607 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  n #  0 )
7877adantl 273 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n #  0 )
7966, 69, 62, 78, 63divcanap7d 8440 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) )  =  ( z  /  n ) )
8079eqeq2d 2111 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) )  <-> 
A  =  ( z  /  n ) ) )
8180biimp3ar 1292 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  A  =  ( (
z  /  ( z  gcd  n ) )  /  ( n  / 
( z  gcd  n
) ) ) )
8276, 81jca 302 . . . . . 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 ) ) ) ) )
8338, 58, 82rspcedvd 2750 . . . . 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 5998 . . . . . . 7  |-  ( x  e.  ( ZZ  X.  NN )  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) ) )
85 elxp6 5998 . . . . . . 7  |-  ( y  e.  ( ZZ  X.  NN )  <->  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )
86 simprl 501 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 1st `  x )  e.  ZZ )
8786ad2antrr 475 . . . . . . . . . . 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 502 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 2nd `  x )  e.  NN )
8988ad2antrr 475 . . . . . . . . . . 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 507 . . . . . . . . . . 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 501 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 1st `  y )  e.  ZZ )
9291ad2antlr 476 . . . . . . . . . . 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 502 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 2nd `  y )  e.  NN )
9493ad2antlr 476 . . . . . . . . . . 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 509 . . . . . . . . . . 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 508 . . . . . . . . . . . 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 510 . . . . . . . . . . . 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 ) ) )
9896, 97eqtr3d 2134 . . . . . . . . . . 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 11570 . . . . . . . . . . 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 ) ) )
10087, 89, 90, 92, 94, 95, 98, 99syl331anc 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
) ) )
101 vex 2644 . . . . . . . . . . . 12  |-  x  e. 
_V
102 1stexg 5996 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
103101, 102ax-mp 7 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
104 2ndexg 5997 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 2nd `  x )  e. 
_V )
105101, 104ax-mp 7 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
106103, 105opth 4097 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. 
<->  ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  =  ( 2nd `  y
) ) )
107100, 106sylibr 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 503 . . . . . . . . 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 505 . . . . . . . . 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 ) >. )
110107, 108, 1093eqtr4d 2142 . . . . . . . 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 )
111110ex 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 ) )
11284, 85, 111syl2anb 287 . . . . . 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 ) )
113112rgen2a 2445 . . . . 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 )
11483, 113jctir 309 . . . 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 ) ) )
1151143expia 1151 . . 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 ) ) ) )
116115rexlimivv 2514 . 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 9264 . 2  |-  ( A  e.  QQ  <->  E. z  e.  ZZ  E. n  e.  NN  A  =  ( z  /  n ) )
118 fveq2 5353 . . . . . 6  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
119 fveq2 5353 . . . . . 6  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
120118, 119oveq12d 5724 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  gcd  ( 2nd `  x ) )  =  ( ( 1st `  y
)  gcd  ( 2nd `  y ) ) )
121120eqeq1d 2108 . . . 4  |-  ( x  =  y  ->  (
( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  <->  ( ( 1st `  y )  gcd  ( 2nd `  y ) )  =  1 ) )
122118, 119oveq12d 5724 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  /  ( 2nd `  x ) )  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) )
123122eqeq2d 2111 . . . 4  |-  ( x  =  y  ->  ( A  =  ( ( 1st `  x )  / 
( 2nd `  x
) )  <->  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )
124121, 123anbi12d 460 . . 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 ) ) ) ) )
125124reu4 2831 . 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 ) ) )
126116, 117, 1253imtr4i 200 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    /\ wa 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    =/= wne 2267   A.wral 2375   E.wrex 2376   E!wreu 2377   _Vcvv 2641   <.cop 3477   class class class wbr 3875    X. cxp 4475   ` cfv 5059  (class class class)co 5706   1stc1st 5967   2ndc2nd 5968   CCcc 7498   RRcr 7499   0cc0 7500   1c1 7501    x. cmul 7505    < clt 7672   # cap 8209    / cdiv 8293   NNcn 8578   NN0cn0 8829   ZZcz 8906   QQcq 9261    || cdvds 11288    gcd cgcd 11430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289  df-gcd 11431
This theorem is referenced by:  qnumdencl  11657  qnumdenbi  11662
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