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Theorem coprimeprodsq 13138
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 10260 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2 nn0z 10260 . . . . . . . 8  |-  ( C  e.  NN0  ->  C  e.  ZZ )
3 gcdcl 12972 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  gcd  C
)  e.  NN0 )
41, 2, 3syl2an 464 . . . . . . 7  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  gcd  C
)  e.  NN0 )
543adant2 976 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  C )  e. 
NN0 )
653ad2ant1 978 . . . . 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 10232 . . . 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 11475 . . 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 989 . . . . . . . . 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 10232 . . . . . . . 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 10187 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  A  e.  CC )
12113ad2ant1 978 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  CC )
13123ad2ant1 978 . . . . . . . 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 9065 . . . . . . 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 962 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  NN0 )
1615nn0cnd 10232 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  CC )
1716sqvald 11475 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( C ^ 2 )  =  ( C  x.  C ) )
1817eqeq1d 2412 . . . . . . . 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 1283 . . . . . . 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 6058 . . . . . 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 987 . . . . . . . 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 10329 . . . . . . 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 10329 . . . . . . 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 13001 . . . . . . 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 1184 . . . . . 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 988 . . . . . . 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 13001 . . . . . . 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 1184 . . . . . 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 2444 . . . . 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 6056 . . . 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 13003 . . . . 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 1184 . . . 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 10329 . . . . 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 12972 . . . . . . . . . 10  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
352, 34sylan 458 . . . . . . . . 9  |-  ( ( C  e.  NN0  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
3635ancoms 440 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  C  e.  NN0 )  -> 
( C  gcd  B
)  e.  NN0 )
37363adant1 975 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e. 
NN0 )
38373ad2ant1 978 . . . . . 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 10329 . . . . 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 13001 . . . . 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 1184 . . . 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 2444 . . 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 980 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  C  e.  ZZ )
44 gcdid 12986 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C  gcd  C )  =  ( abs `  C
) )
4543, 44syl 16 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  C )  =  ( abs `  C
) )
4645oveq1d 6055 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( ( abs `  C )  gcd  B
) )
47 simp2 958 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  B  e.  ZZ )
48 gcdabs1 12989 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
4943, 47, 48syl2anc 643 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
5046, 49eqtrd 2436 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  B ) )
51 gcdass 13000 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
5243, 43, 47, 51syl3anc 1184 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
53 gcdcom 12975 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  =  ( B  gcd  C ) )
5443, 47, 53syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  =  ( B  gcd  C
) )
5550, 52, 543eqtr3d 2444 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  ( C  gcd  B ) )  =  ( B  gcd  C ) )
5655oveq2d 6056 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  ( C  gcd  ( C  gcd  B ) ) )  =  ( A  gcd  ( B  gcd  C ) ) )
5713ad2ant1 978 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  ZZ )
5837nn0zd 10329 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e.  ZZ )
59 gcdass 13000 . . . . . . . . . 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 1184 . . . . . . . . 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 13000 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6257, 47, 43, 61syl3anc 1184 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6356, 60, 623eqtr4d 2446 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( ( A  gcd  B )  gcd  C ) )
6463eqeq1d 2412 . . . . . . 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 472 . . . . . 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 6056 . . . . 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 977 . . . 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 9061 . . . 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 2436 . . 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 2441 . 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 1155 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    x. cmul 8951   2c2 10005   NN0cn0 10177   ZZcz 10238   ^cexp 11337   abscabs 11994    gcd cgcd 12961
This theorem is referenced by:  coprimeprodsq2  13139  pythagtriplem6  13150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962
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