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Theorem coprimeprodsq 12878
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 10062 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2 nn0z 10062 . . . . . . . 8  |-  ( C  e.  NN0  ->  C  e.  ZZ )
3 gcdcl 12712 . . . . . . . 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 10036 . . . 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 11258 . . 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 10036 . . . . . . . 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 9991 . . . . . . . . . 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 8872 . . . . . . 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 10036 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  CC )
1716sqvald 11258 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( C ^ 2 )  =  ( C  x.  C ) )
1817eqeq1d 2304 . . . . . . . 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 5892 . . . . . 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 10131 . . . . . . 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 10131 . . . . . . 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 12741 . . . . . . 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 12741 . . . . . . 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 2336 . . . . 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 5890 . . . 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 12743 . . . . 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 10131 . . . . 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 12712 . . . . . . . . . 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 10131 . . . . 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 12741 . . . . 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 2336 . . 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 12726 . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . 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 12729 . . . . . . . . . . . . 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 2328 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  B ) )
51 gcdass 12740 . . . . . . . . . . . 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 12715 . . . . . . . . . . . 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 2336 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  ( C  gcd  B ) )  =  ( B  gcd  C ) )
5655oveq2d 5890 . . . . . . . . 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 10131 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e.  ZZ )
59 gcdass 12740 . . . . . . . . . 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 12740 . . . . . . . . . 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 2338 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( ( A  gcd  B )  gcd  C ) )
6463eqeq1d 2304 . . . . . . 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 5890 . . . . 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 8868 . . . 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 2328 . . 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 2333 . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    x. cmul 8758   2c2 9811   NN0cn0 9981   ZZcz 10040   ^cexp 11120   abscabs 11735    gcd cgcd 12701
This theorem is referenced by:  coprimeprodsq2  12879  pythagtriplem6  12890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702
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