ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  qredeu Unicode version

Theorem qredeu 10985
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 8705 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  ZZ )
2 gcddvds 10861 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( ( z  gcd  n )  ||  z  /\  ( z  gcd  n
)  ||  n )
)
32simpld 110 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  z )
41, 3sylan2 280 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  z )
5 gcdcl 10864 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  e.  NN0 )
61, 5sylan2 280 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN0 )
76nn0zd 8802 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  ZZ )
8 simpl 107 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  ZZ )
91adantl 271 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  ZZ )
10 nnne0 8388 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  n  =/=  0 )
1110neneqd 2272 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  -.  n  =  0 )
1211intnand 876 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  -.  ( z  =  0  /\  n  =  0 ) )
1312adantl 271 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  -.  ( z  =  0  /\  n  =  0 ) )
14 gcdn0cl 10860 . . . . . . . . . . . 12  |-  ( ( ( z  e.  ZZ  /\  n  e.  ZZ )  /\  -.  ( z  =  0  /\  n  =  0 ) )  ->  ( z  gcd  n )  e.  NN )
158, 9, 13, 14syl21anc 1171 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  NN )
1615nnne0d 8404 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  =/=  0 )
17 dvdsval2 10705 . . . . . . . . . 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 1172 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  z  <->  ( z  /  ( z  gcd  n ) )  e.  ZZ ) )
194, 18mpbid 145 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  /  (
z  gcd  n )
)  e.  ZZ )
20193adant3 961 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  ZZ )
212simprd 112 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  ZZ )  ->  ( z  gcd  n
)  ||  n )
221, 21sylan2 280 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  ||  n )
23 dvdsval2 10705 . . . . . . . . . . . 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 1172 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  ||  n  <->  ( n  /  ( z  gcd  n ) )  e.  ZZ ) )
2522, 24mpbid 145 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( n  /  (
z  gcd  n )
)  e.  ZZ )
26 nnre 8367 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  e.  RR )
2726adantl 271 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  RR )
286nn0red 8663 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  RR )
29 nngt0 8385 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  0  <  n )
3029adantl 271 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  n )
3115nngt0d 8403 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( z  gcd  n ) )
3227, 28, 30, 31divgt0d 8334 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  0  <  ( n  /  ( z  gcd  n ) ) )
3325, 32jca 300 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
34333adant3 961 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( n  / 
( z  gcd  n
) )  e.  ZZ  /\  0  <  ( n  /  ( z  gcd  n ) ) ) )
35 elnnz 8696 . . . . . . . 8  |-  ( ( n  /  ( z  gcd  n ) )  e.  NN  <->  ( (
n  /  ( z  gcd  n ) )  e.  ZZ  /\  0  <  ( n  /  (
z  gcd  n )
) ) )
3634, 35sylibr 132 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  NN )
37 opelxpi 4444 . . . . . . 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 403 . . . . . 6  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  <. ( z  /  (
z  gcd  n )
) ,  ( n  /  ( z  gcd  n ) ) >.  e.  ( ZZ  X.  NN ) )
39 fveq2 5270 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 1st `  x
)  =  ( 1st `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
40 simp1 941 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
z  e.  ZZ )
41153adant3 961 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  gcd  n
)  e.  NN )
42 znq 9044 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( z  / 
( z  gcd  n
) )  e.  QQ )
4340, 41, 42syl2anc 403 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( z  /  (
z  gcd  n )
)  e.  QQ )
4493adant3 961 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  n  e.  ZZ )
45 znq 9044 . . . . . . . . . . . 12  |-  ( ( n  e.  ZZ  /\  ( z  gcd  n
)  e.  NN )  ->  ( n  / 
( z  gcd  n
) )  e.  QQ )
4644, 41, 45syl2anc 403 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( n  /  (
z  gcd  n )
)  e.  QQ )
47 op1stg 5880 . . . . . . . . . . 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 403 . . . . . . . . . 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 2139 . . . . . . . . 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 5270 . . . . . . . . . 10  |-  ( x  =  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. ( z  / 
( z  gcd  n
) ) ,  ( n  /  ( z  gcd  n ) )
>. ) )
51 op2ndg 5881 . . . . . . . . . . 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 403 . . . . . . . . . 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 2139 . . . . . . . . 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 5633 . . . . . . . 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 2093 . . . . . . 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 5633 . . . . . . . 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 2096 . . . . . . 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 457 . . . . . 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 10866 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  NN0 )
6059nn0cnd 8664 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  e.  CC )
61 1cnd 7451 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  1  e.  CC )
626nn0cnd 8664 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
)  e.  CC )
6315nnap0d 8405 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( z  gcd  n
) #  0 )
6462mulid1d 7452 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  1 )  =  ( z  gcd  n ) )
65 zcn 8691 . . . . . . . . . . . . 13  |-  ( z  e.  ZZ  ->  z  e.  CC )
6665adantr 270 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  z  e.  CC )
6766, 62, 63divcanap2d 8200 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
z  /  ( z  gcd  n ) ) )  =  z )
68 nncn 8368 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  n  e.  CC )
6968adantl 271 . . . . . . . . . . . 12  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n  e.  CC )
7069, 62, 63divcanap2d 8200 . . . . . . . . . . 11  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  gcd  n )  x.  (
n  /  ( z  gcd  n ) ) )  =  n )
7167, 70oveq12d 5633 . . . . . . . . . 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 10911 . . . . . . . . . . 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 1172 . . . . . . . . . 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 2124 . . . . . . . . 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 8074 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
76753adant3 961 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  -> 
( ( z  / 
( z  gcd  n
) )  gcd  (
n  /  ( z  gcd  n ) ) )  =  1 )
77 nnap0 8389 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  n #  0 )
7877adantl 271 . . . . . . . . . 10  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  n #  0 )
7966, 69, 62, 78, 63divcanap7d 8226 . . . . . . . . 9  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( ( z  / 
( z  gcd  n
) )  /  (
n  /  ( z  gcd  n ) ) )  =  ( z  /  n ) )
8079eqeq2d 2096 . . . . . . . 8  |-  ( ( z  e.  ZZ  /\  n  e.  NN )  ->  ( A  =  ( ( z  /  (
z  gcd  n )
)  /  ( n  /  ( z  gcd  n ) ) )  <-> 
A  =  ( z  /  n ) ) )
8180biimp3ar 1280 . . . . . . 7  |-  ( ( z  e.  ZZ  /\  n  e.  NN  /\  A  =  ( z  /  n ) )  ->  A  =  ( (
z  /  ( z  gcd  n ) )  /  ( n  / 
( z  gcd  n
) ) ) )
8276, 81jca 300 . . . . . 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 2721 . . . . 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 5899 . . . . . . 7  |-  ( x  e.  ( ZZ  X.  NN )  <->  ( x  = 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) ) )
85 elxp6 5899 . . . . . . 7  |-  ( y  e.  ( ZZ  X.  NN )  <->  ( y  = 
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  /\  (
( 1st `  y
)  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) ) )
86 simprl 498 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 1st `  x )  e.  ZZ )
8786ad2antrr 472 . . . . . . . . . . 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 499 . . . . . . . . . . . 12  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  ZZ  /\  ( 2nd `  x )  e.  NN ) )  ->  ( 2nd `  x )  e.  NN )
8988ad2antrr 472 . . . . . . . . . . 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 504 . . . . . . . . . . 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 498 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 1st `  y )  e.  ZZ )
9291ad2antlr 473 . . . . . . . . . . 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 499 . . . . . . . . . . . 12  |-  ( ( y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  /\  ( ( 1st `  y )  e.  ZZ  /\  ( 2nd `  y )  e.  NN ) )  ->  ( 2nd `  y )  e.  NN )
9493ad2antlr 473 . . . . . . . . . . 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 506 . . . . . . . . . . 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 505 . . . . . . . . . . . 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 507 . . . . . . . . . . . 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 2119 . . . . . . . . . . 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 10984 . . . . . . . . . . 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 1197 . . . . . . . . . 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 2618 . . . . . . . . . . . 12  |-  x  e. 
_V
102 1stexg 5897 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 1st `  x )  e. 
_V )
103101, 102ax-mp 7 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
104 2ndexg 5898 . . . . . . . . . . . 12  |-  ( x  e.  _V  ->  ( 2nd `  x )  e. 
_V )
105101, 104ax-mp 7 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
106103, 105opth 4040 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. 
<->  ( ( 1st `  x
)  =  ( 1st `  y )  /\  ( 2nd `  x )  =  ( 2nd `  y
) ) )
107100, 106sylibr 132 . . . . . . . . 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 500 . . . . . . . . 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 502 . . . . . . . . 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 2127 . . . . . . . 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 113 . . . . . . 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 285 . . . . . 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 2425 . . . . 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 306 . . . 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 1143 . . 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 2490 . 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 9042 . 2  |-  ( A  e.  QQ  <->  E. z  e.  ZZ  E. n  e.  NN  A  =  ( z  /  n ) )
118 fveq2 5270 . . . . . 6  |-  ( x  =  y  ->  ( 1st `  x )  =  ( 1st `  y
) )
119 fveq2 5270 . . . . . 6  |-  ( x  =  y  ->  ( 2nd `  x )  =  ( 2nd `  y
) )
120118, 119oveq12d 5633 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  gcd  ( 2nd `  x ) )  =  ( ( 1st `  y
)  gcd  ( 2nd `  y ) ) )
121120eqeq1d 2093 . . . 4  |-  ( x  =  y  ->  (
( ( 1st `  x
)  gcd  ( 2nd `  x ) )  =  1  <->  ( ( 1st `  y )  gcd  ( 2nd `  y ) )  =  1 ) )
122118, 119oveq12d 5633 . . . . 5  |-  ( x  =  y  ->  (
( 1st `  x
)  /  ( 2nd `  x ) )  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) )
123122eqeq2d 2096 . . . 4  |-  ( x  =  y  ->  ( A  =  ( ( 1st `  x )  / 
( 2nd `  x
) )  <->  A  =  ( ( 1st `  y
)  /  ( 2nd `  y ) ) ) )
124121, 123anbi12d 457 . . 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 2800 . 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 199 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 102    <-> wb 103    /\ w3a 922    = wceq 1287    e. wcel 1436    =/= wne 2251   A.wral 2355   E.wrex 2356   E!wreu 2357   _Vcvv 2615   <.cop 3434   class class class wbr 3822    X. cxp 4411   ` cfv 4983  (class class class)co 5615   1stc1st 5868   2ndc2nd 5869   CCcc 7295   RRcr 7296   0cc0 7297   1c1 7298    x. cmul 7302    < clt 7469   # cap 8002    / cdiv 8081   NNcn 8360   NN0cn0 8609   ZZcz 8686   QQcq 9039    || cdvds 10702    gcd cgcd 10844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-1re 7386  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-mulrcl 7391  ax-addcom 7392  ax-mulcom 7393  ax-addass 7394  ax-mulass 7395  ax-distr 7396  ax-i2m1 7397  ax-0lt1 7398  ax-1rid 7399  ax-0id 7400  ax-rnegex 7401  ax-precex 7402  ax-cnre 7403  ax-pre-ltirr 7404  ax-pre-ltwlin 7405  ax-pre-lttrn 7406  ax-pre-apti 7407  ax-pre-ltadd 7408  ax-pre-mulgt0 7409  ax-pre-mulext 7410  ax-arch 7411  ax-caucvg 7412
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-ilim 4172  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-frec 6112  df-sup 6626  df-pnf 7471  df-mnf 7472  df-xr 7473  df-ltxr 7474  df-le 7475  df-sub 7602  df-neg 7603  df-reap 7996  df-ap 8003  df-div 8082  df-inn 8361  df-2 8419  df-3 8420  df-4 8421  df-n0 8610  df-z 8687  df-uz 8955  df-q 9040  df-rp 9070  df-fz 9360  df-fzo 9485  df-fl 9608  df-mod 9661  df-iseq 9783  df-iexp 9857  df-cj 10175  df-re 10176  df-im 10177  df-rsqrt 10330  df-abs 10331  df-dvds 10703  df-gcd 10845
This theorem is referenced by:  qnumdencl  11071  qnumdenbi  11076
  Copyright terms: Public domain W3C validator