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Theorem coprimeprodsq 12862
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 10046 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2 nn0z 10046 . . . . . . . 8  |-  ( C  e.  NN0  ->  C  e.  ZZ )
3 gcdcl 12696 . . . . . . . 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 10020 . . . 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 11242 . . 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 10020 . . . . . . . 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 9975 . . . . . . . . . 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 8856 . . . . . . 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 10020 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  CC )
1716sqvald 11242 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( C ^ 2 )  =  ( C  x.  C ) )
1817eqeq1d 2291 . . . . . . . 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 5876 . . . . . 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 10115 . . . . . . 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 10115 . . . . . . 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 12725 . . . . . . 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 12725 . . . . . . 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 2323 . . . . 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 5874 . . . 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 12727 . . . . 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 10115 . . . . 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 12696 . . . . . . . . . 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 10115 . . . . 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 12725 . . . . 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 2323 . . 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 12710 . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . 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 12713 . . . . . . . . . . . . 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 2315 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  B ) )
51 gcdass 12724 . . . . . . . . . . . 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 12699 . . . . . . . . . . . 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 2323 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  ( C  gcd  B ) )  =  ( B  gcd  C ) )
5655oveq2d 5874 . . . . . . . . 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 10115 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e.  ZZ )
59 gcdass 12724 . . . . . . . . . 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 12724 . . . . . . . . . 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 2325 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( ( A  gcd  B )  gcd  C ) )
6463eqeq1d 2291 . . . . . . 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 5874 . . . . 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 8852 . . . 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 2315 . . 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 2320 . 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 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104   abscabs 11719    gcd cgcd 12685
This theorem is referenced by:  coprimeprodsq2  12863  pythagtriplem6  12874
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686
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