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Theorem qredeu 11705
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 9041 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
2 gcddvds 11579 . . . . . . . . . . 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 284 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  z )
5 gcdcl 11582 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  e.  NN0 )
61, 5sylan2 284 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN0 )
76nn0zd 9139 . . . . . . . . . 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 275 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  ZZ )
10 nnne0 8716 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
1110neneqd 2306 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  -.  n  =  0 )
1211intnand 901 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  -.  ( z  =  0  /\  n  =  0 ) )
1312adantl 275 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  -.  ( z  =  0  /\  n  =  0 ) )
14 gcdn0cl 11578 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ZZ  /\  n  e.  ZZ )  /\  -.  ( z  =  0  /\  n  =  0 ) )  ->  ( z  gcd  n )  e.  NN )
158, 9, 13, 14syl21anc 1200 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN )
1615nnne0d 8733 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  =/=  0 )
17 dvdsval2 11423 . . . . . . . . . 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 1201 . . . . . . . . 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 986 . . . . . . 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 284 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  n )
23 dvdsval2 11423 . . . . . . . . . . . 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 1201 . . . . . . . . . . 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 8695 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  RR )
2726adantl 275 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  RR )
286nn0red 8999 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  RR )
29 nngt0 8713 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  0  <  n )
3029adantl 275 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  n )
3115nngt0d 8732 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( z  gcd  n ) )
3227, 28, 30, 31divgt0d 8661 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( n  /  ( z  gcd  n ) ) )
3325, 32jca 304 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
34333adant3 986 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
35 elnnz 9032 . . . . . . . 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 4541 . . . . . . 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 408 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  <. ( z  /  (
z  gcd  n )
) ,  ( n  /  ( z  gcd  n ) ) >.  e.  ( ZZ  X.  NN ) )
39 fveq2 5389 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 1st `  x
)  =  ( 1st `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
40 simp1 966 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
z  e.  ZZ )
41153adant3 986 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  gcd  n
)  e.  NN )
42 znq 9384 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( z  / 
( z  gcd  n
) )  e.  QQ )
4340, 41, 42syl2anc 408 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  QQ )
4493adant3 986 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  n  e.  ZZ )
45 znq 9384 . . . . . . . . . . . 12  |-  ( ( n  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( n  / 
( z  gcd  n
) )  e.  QQ )
4644, 41, 45syl2anc 408 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  QQ )
47 op1stg 6016 . . . . . . . . . . 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 408 . . . . . . . . . 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 2172 . . . . . . . . 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 5389 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
51 op2ndg 6017 . . . . . . . . . . 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 408 . . . . . . . . . 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 2172 . . . . . . . . 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 5760 . . . . . . . 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 2126 . . . . . . 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 5760 . . . . . . . 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 2129 . . . . . . 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 464 . . . . . 6  |-  ( ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  /\  x  =  <. ( z  /  ( z  gcd  n ) ) ,  ( n  /  (
z  gcd  n )
) >. )  ->  (
( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) )  <->  ( ( ( z  /  ( z  gcd  n ) )  gcd  ( n  / 
( z  gcd  n
) ) )  =  1  /\  A  =  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) ) ) ) )
5919, 25gcdcld 11584 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  NN0 )
6059nn0cnd 9000 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  CC )
61 1cnd 7750 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  1  e.  CC )
626nn0cnd 9000 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  CC )
6315nnap0d 8734 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
) #  0 )
6462mulid1d 7751 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  1 )  =  ( z  gcd  n ) )
65 zcn 9027 . . . . . . . . . . . . 13  |-  ( z  e.  ZZ  ->  z  e.  CC )
6665adantr 274 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  CC )
6766, 62, 63divcanap2d 8520 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
z  /  ( z  gcd  n ) ) )  =  z )
68 nncn 8696 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  CC )
6968adantl 275 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  CC )
7069, 62, 63divcanap2d 8520 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
n  /  ( z  gcd  n ) ) )  =  n )
7167, 70oveq12d 5760 . . . . . . . . . 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 11631 . . . . . . . . . . 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 1201 . . . . . . . . . 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 2157 . . . . . . . . 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 8392 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
76753adant3 986 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
77 nnap0 8717 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  n #  0 )
7877adantl 275 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n #  0 )
7966, 69, 62, 78, 63divcanap7d 8547 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) )  =  ( z  /  n ) )
8079eqeq2d 2129 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) )  <-> 
A  =  ( z  /  n ) ) )
8180biimp3ar 1309 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  A  =  ( (
z  /  ( z  gcd  n ) )  /  ( n  / 
( z  gcd  n
) ) ) )
8276, 81jca 304 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( ( z  /  ( z  gcd  n ) )  gcd  ( n  /  (
z  gcd  n )
) )  =  1  /\  A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) ) ) )
8338, 58, 82rspcedvd 2769 . . . . 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 6035 . . . . . . 7  |-  ( x  e.  ( ZZ  X.  NN )  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) ) )
85 elxp6 6035 . . . . . . 7  |-  ( y  e.  ( ZZ  X.  NN )  <->  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )
86 simprl 505 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 1st `  x )  e.  ZZ )
8786ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  /\  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )  /\  ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) ) )  ->  ( 1st `  x )  e.  ZZ )
88 simprr 506 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 2nd `  x )  e.  NN )
8988ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  /\  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )  /\  ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) ) )  ->  ( 2nd `  x )  e.  NN )
90 simprll 511 . . . . . . . . . . 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 505 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 1st `  y )  e.  ZZ )
9291ad2antlr 480 . . . . . . . . . . 11  |-  ( ( ( ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  /\  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )  /\  ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) ) )  ->  ( 1st `  y )  e.  ZZ )
93 simprr 506 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 2nd `  y )  e.  NN )
9493ad2antlr 480 . . . . . . . . . . 11  |-  ( ( ( ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  /\  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )  /\  ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) ) )  ->  ( 2nd `  y )  e.  NN )
95 simprrl 513 . . . . . . . . . . 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 512 . . . . . . . . . . . 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 514 . . . . . . . . . . . 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 2152 . . . . . . . . . . 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 11704 . . . . . . . . . . 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 1226 . . . . . . . . . 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 2663 . . . . . . . . . . . 12  |-  x  e. 
_V
102 1stexg 6033 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
103101, 102ax-mp 5 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
104 2ndexg 6034 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 2nd `  x )  e. 
_V )
105101, 104ax-mp 5 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
106103, 105opth 4129 . . . . . . . . . 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 507 . . . . . . . . 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 509 . . . . . . . . 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 2160 . . . . . . . 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 289 . . . . . 6  |-  ( ( x  e.  ( ZZ 
X.  NN )  /\  y  e.  ( ZZ  X.  NN ) )  -> 
( ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )  ->  x  =  y ) )
113112rgen2a 2463 . . . . 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 311 . . . 4  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( E. x  e.  ( ZZ  X.  NN ) ( ( ( 1st `  x )  gcd  ( 2nd `  x
) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  A. x  e.  ( ZZ  X.  NN ) A. y  e.  ( ZZ  X.  NN ) ( ( ( ( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x
)  /  ( 2nd `  x ) ) )  /\  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )  ->  x  =  y ) ) )
1151143expia 1168 . . 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 2532 . 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 9382 . 2  |-  ( A  e.  QQ  <->  E. z  e.  ZZ  E. n  e.  NN  A  =  ( z  /  n ) )
118 fveq2 5389 . . . . . 6  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
119 fveq2 5389 . . . . . 6  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
120118, 119oveq12d 5760 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  gcd  ( 2nd `  x ) )  =  ( ( 1st `  y
)  gcd  ( 2nd `  y ) ) )
121120eqeq1d 2126 . . . 4  |-  ( x  =  y  ->  (
( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  <->  ( ( 1st `  y )  gcd  ( 2nd `  y ) )  =  1 ) )
122118, 119oveq12d 5760 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  /  ( 2nd `  x ) )  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) )
123122eqeq2d 2129 . . . 4  |-  ( x  =  y  ->  ( A  =  ( ( 1st `  x )  / 
( 2nd `  x
) )  <->  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )
124121, 123anbi12d 464 . . 3  |-  ( x  =  y  ->  (
( ( ( 1st `  x )  gcd  ( 2nd `  x ) )  =  1  /\  A  =  ( ( 1st `  x )  /  ( 2nd `  x ) ) )  <->  ( ( ( 1st `  y )  gcd  ( 2nd `  y
) )  =  1  /\  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) ) )
125124reu4 2851 . 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 947    = wceq 1316    e. wcel 1465    =/= wne 2285   A.wral 2393   E.wrex 2394   E!wreu 2395   _Vcvv 2660   <.cop 3500   class class class wbr 3899    X. cxp 4507   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    x. cmul 7593    < clt 7768   # cap 8311    / cdiv 8400   NNcn 8688   NN0cn0 8945   ZZcz 9022   QQcq 9379    || cdvds 11420    gcd cgcd 11562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-fl 10011  df-mod 10064  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-dvds 11421  df-gcd 11563
This theorem is referenced by:  qnumdencl  11792  qnumdenbi  11797
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