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Theorem coprimeprodsq 12858
Description: If three numbers are coprime, and the square of one is the product of the other two, then there is a formula for the other two in terms of  gcd and square. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
coprimeprodsq  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  ->  A  =  ( ( A  gcd  C
) ^ 2 ) ) )

Proof of Theorem coprimeprodsq
StepHypRef Expression
1 nn0z 10042 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2 nn0z 10042 . . . . . . . 8  |-  ( C  e.  NN0  ->  C  e.  ZZ )
3 gcdcl 12692 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  gcd  C
)  e.  NN0 )
41, 2, 3syl2an 463 . . . . . . 7  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  gcd  C
)  e.  NN0 )
543adant2 974 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  C )  e. 
NN0 )
653ad2ant1 976 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  gcd  C )  e. 
NN0 )
76nn0cnd 10016 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  gcd  C )  e.  CC )
87sqvald 11238 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  gcd  C
) ^ 2 )  =  ( ( A  gcd  C )  x.  ( A  gcd  C
) ) )
9 simp13 987 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  C  e.  NN0 )
109nn0cnd 10016 . . . . . . . 8  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  C  e.  CC )
11 nn0cn 9971 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  A  e.  CC )
12113ad2ant1 976 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  CC )
13123ad2ant1 976 . . . . . . . 8  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  A  e.  CC )
1410, 13mulcomd 8852 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( C  x.  A )  =  ( A  x.  C ) )
15 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  NN0 )
1615nn0cnd 10016 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  CC )
1716sqvald 11238 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( C ^ 2 )  =  ( C  x.  C ) )
1817eqeq1d 2292 . . . . . . . 8  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  <-> 
( C  x.  C
)  =  ( A  x.  B ) ) )
1918biimp3a 1281 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( C  x.  C )  =  ( A  x.  B ) )
2014, 19oveq12d 5838 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( C  x.  A
)  gcd  ( C  x.  C ) )  =  ( ( A  x.  C )  gcd  ( A  x.  B )
) )
21 simp11 985 . . . . . . . 8  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  A  e.  NN0 )
2221nn0zd 10111 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  A  e.  ZZ )
239nn0zd 10111 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  C  e.  ZZ )
24 mulgcd 12721 . . . . . . 7  |-  ( ( C  e.  NN0  /\  A  e.  ZZ  /\  C  e.  ZZ )  ->  (
( C  x.  A
)  gcd  ( C  x.  C ) )  =  ( C  x.  ( A  gcd  C ) ) )
259, 22, 23, 24syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( C  x.  A
)  gcd  ( C  x.  C ) )  =  ( C  x.  ( A  gcd  C ) ) )
26 simp12 986 . . . . . . 7  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  B  e.  ZZ )
27 mulgcd 12721 . . . . . . 7  |-  ( ( A  e.  NN0  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  x.  C
)  gcd  ( A  x.  B ) )  =  ( A  x.  ( C  gcd  B ) ) )
2821, 23, 26, 27syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  x.  C
)  gcd  ( A  x.  B ) )  =  ( A  x.  ( C  gcd  B ) ) )
2920, 25, 283eqtr3d 2324 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( C  x.  ( A  gcd  C ) )  =  ( A  x.  ( C  gcd  B ) ) )
3029oveq2d 5836 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  x.  ( A  gcd  C ) )  gcd  ( C  x.  ( A  gcd  C ) ) )  =  ( ( A  x.  ( A  gcd  C ) )  gcd  ( A  x.  ( C  gcd  B ) ) ) )
31 mulgcdr 12723 . . . . 5  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  ( A  gcd  C )  e. 
NN0 )  ->  (
( A  x.  ( A  gcd  C ) )  gcd  ( C  x.  ( A  gcd  C ) ) )  =  ( ( A  gcd  C
)  x.  ( A  gcd  C ) ) )
3222, 23, 6, 31syl3anc 1182 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  x.  ( A  gcd  C ) )  gcd  ( C  x.  ( A  gcd  C ) ) )  =  ( ( A  gcd  C
)  x.  ( A  gcd  C ) ) )
336nn0zd 10111 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  gcd  C )  e.  ZZ )
34 gcdcl 12692 . . . . . . . . . 10  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
352, 34sylan 457 . . . . . . . . 9  |-  ( ( C  e.  NN0  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
3635ancoms 439 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  C  e.  NN0 )  -> 
( C  gcd  B
)  e.  NN0 )
37363adant1 973 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e. 
NN0 )
38373ad2ant1 976 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( C  gcd  B )  e. 
NN0 )
3938nn0zd 10111 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( C  gcd  B )  e.  ZZ )
40 mulgcd 12721 . . . . 5  |-  ( ( A  e.  NN0  /\  ( A  gcd  C )  e.  ZZ  /\  ( C  gcd  B )  e.  ZZ )  ->  (
( A  x.  ( A  gcd  C ) )  gcd  ( A  x.  ( C  gcd  B ) ) )  =  ( A  x.  ( ( A  gcd  C )  gcd  ( C  gcd  B ) ) ) )
4121, 33, 39, 40syl3anc 1182 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  x.  ( A  gcd  C ) )  gcd  ( A  x.  ( C  gcd  B ) ) )  =  ( A  x.  ( ( A  gcd  C )  gcd  ( C  gcd  B ) ) ) )
4230, 32, 413eqtr3d 2324 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  (
( A  gcd  C
)  x.  ( A  gcd  C ) )  =  ( A  x.  ( ( A  gcd  C )  gcd  ( C  gcd  B ) ) ) )
4323ad2ant3 978 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  C  e.  ZZ )
44 gcdid 12706 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C  gcd  C )  =  ( abs `  C
) )
4543, 44syl 15 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  C )  =  ( abs `  C
) )
4645oveq1d 5835 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( ( abs `  C )  gcd  B
) )
47 simp2 956 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  B  e.  ZZ )
48 gcdabs1 12709 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
4943, 47, 48syl2anc 642 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
5046, 49eqtrd 2316 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  B ) )
51 gcdass 12720 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
5243, 43, 47, 51syl3anc 1182 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
53 gcdcom 12695 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  =  ( B  gcd  C ) )
5443, 47, 53syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  =  ( B  gcd  C
) )
5550, 52, 543eqtr3d 2324 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  ( C  gcd  B ) )  =  ( B  gcd  C ) )
5655oveq2d 5836 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  ( C  gcd  ( C  gcd  B ) ) )  =  ( A  gcd  ( B  gcd  C ) ) )
5713ad2ant1 976 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  ZZ )
5837nn0zd 10111 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e.  ZZ )
59 gcdass 12720 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ  /\  ( C  gcd  B )  e.  ZZ )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( A  gcd  ( C  gcd  ( C  gcd  B ) ) ) )
6057, 43, 58, 59syl3anc 1182 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( A  gcd  ( C  gcd  ( C  gcd  B ) ) ) )
61 gcdass 12720 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6257, 47, 43, 61syl3anc 1182 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6356, 60, 623eqtr4d 2326 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( ( A  gcd  B )  gcd  C ) )
6463eqeq1d 2292 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( ( A  gcd  C )  gcd  ( C  gcd  B ) )  =  1  <->  ( ( A  gcd  B )  gcd 
C )  =  1 ) )
6564biimpar 471 . . . . . 6  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( A  gcd  C )  gcd  ( C  gcd  B ) )  =  1 )
6665oveq2d 5836 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( A  x.  (
( A  gcd  C
)  gcd  ( C  gcd  B ) ) )  =  ( A  x.  1 ) )
67663adant3 975 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  x.  ( ( A  gcd  C )  gcd  ( C  gcd  B
) ) )  =  ( A  x.  1 ) )
6813mulid1d 8848 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  x.  1 )  =  A )
6967, 68eqtrd 2316 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  ( A  x.  ( ( A  gcd  C )  gcd  ( C  gcd  B
) ) )  =  A )
708, 42, 693eqtrrd 2321 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1  /\  ( C ^ 2 )  =  ( A  x.  B
) )  ->  A  =  ( ( A  gcd  C ) ^
2 ) )
71703expia 1153 1  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( ( C ^
2 )  =  ( A  x.  B )  ->  A  =  ( ( A  gcd  C
) ^ 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   ` cfv 5221  (class class class)co 5820   CCcc 8731   1c1 8734    x. cmul 8738   2c2 9791   NN0cn0 9961   ZZcz 10020   ^cexp 11100   abscabs 11715    gcd cgcd 12681
This theorem is referenced by:  coprimeprodsq2  12859  pythagtriplem6  12870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-dvds 12528  df-gcd 12682
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