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Theorem coprimeprodsq 12825
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 10014 . . . . . . . 8  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2 nn0z 10014 . . . . . . . 8  |-  ( C  e.  NN0  ->  C  e.  ZZ )
3 gcdcl 12659 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  gcd  C
)  e.  NN0 )
41, 2, 3syl2an 465 . . . . . . 7  |-  ( ( A  e.  NN0  /\  C  e.  NN0 )  -> 
( A  gcd  C
)  e.  NN0 )
543adant2 979 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  C )  e. 
NN0 )
653ad2ant1 981 . . . . 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 9988 . . . 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 11209 . . 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 992 . . . . . . . . 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 9988 . . . . . . . 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 9943 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  A  e.  CC )
12113ad2ant1 981 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  CC )
13123ad2ant1 981 . . . . . . . 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 8824 . . . . . . 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 965 . . . . . . . . . . 11  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  NN0 )
1615nn0cnd 9988 . . . . . . . . . 10  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  ->  C  e.  CC )
1716sqvald 11209 . . . . . . . . 9  |-  ( ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  /\  ( ( A  gcd  B )  gcd  C )  =  1 )  -> 
( C ^ 2 )  =  ( C  x.  C ) )
1817eqeq1d 2266 . . . . . . . 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 1286 . . . . . . 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 5810 . . . . . 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 990 . . . . . . . 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 10083 . . . . . . 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 10083 . . . . . . 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 12688 . . . . . . 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 1187 . . . . . 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 991 . . . . . . 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 12688 . . . . . . 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 1187 . . . . . 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 2298 . . . . 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 5808 . . . 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 12690 . . . . 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 1187 . . . 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 10083 . . . . 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 12659 . . . . . . . . . 10  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
352, 34sylan 459 . . . . . . . . 9  |-  ( ( C  e.  NN0  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  e.  NN0 )
3635ancoms 441 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  C  e.  NN0 )  -> 
( C  gcd  B
)  e.  NN0 )
37363adant1 978 . . . . . . 7  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e. 
NN0 )
38373ad2ant1 981 . . . . . 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 10083 . . . . 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 12688 . . . . 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 1187 . . . 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 2298 . . 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 983 . . . . . . . . . . . . . 14  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  C  e.  ZZ )
44 gcdid 12673 . . . . . . . . . . . . . 14  |-  ( C  e.  ZZ  ->  ( C  gcd  C )  =  ( abs `  C
) )
4543, 44syl 17 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  C )  =  ( abs `  C
) )
4645oveq1d 5807 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( ( abs `  C )  gcd  B
) )
47 simp2 961 . . . . . . . . . . . . 13  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  B  e.  ZZ )
48 gcdabs1 12676 . . . . . . . . . . . . 13  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
4943, 47, 48syl2anc 645 . . . . . . . . . . . 12  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( abs `  C
)  gcd  B )  =  ( C  gcd  B ) )
5046, 49eqtrd 2290 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  B ) )
51 gcdass 12687 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  C  e.  ZZ  /\  B  e.  ZZ )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
5243, 43, 47, 51syl3anc 1187 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( C  gcd  C
)  gcd  B )  =  ( C  gcd  ( C  gcd  B ) ) )
53 gcdcom 12662 . . . . . . . . . . . 12  |-  ( ( C  e.  ZZ  /\  B  e.  ZZ )  ->  ( C  gcd  B
)  =  ( B  gcd  C ) )
5443, 47, 53syl2anc 645 . . . . . . . . . . 11  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  =  ( B  gcd  C
) )
5550, 52, 543eqtr3d 2298 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  ( C  gcd  B ) )  =  ( B  gcd  C ) )
5655oveq2d 5808 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( A  gcd  ( C  gcd  ( C  gcd  B ) ) )  =  ( A  gcd  ( B  gcd  C ) ) )
5713ad2ant1 981 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  A  e.  ZZ )
5837nn0zd 10083 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  ( C  gcd  B )  e.  ZZ )
59 gcdass 12687 . . . . . . . . . 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 1187 . . . . . . . . 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 12687 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6257, 47, 43, 61syl3anc 1187 . . . . . . . . 9  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  B
)  gcd  C )  =  ( A  gcd  ( B  gcd  C ) ) )
6356, 60, 623eqtr4d 2300 . . . . . . . 8  |-  ( ( A  e.  NN0  /\  B  e.  ZZ  /\  C  e.  NN0 )  ->  (
( A  gcd  C
)  gcd  ( C  gcd  B ) )  =  ( ( A  gcd  B )  gcd  C ) )
6463eqeq1d 2266 . . . . . . 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 473 . . . . . 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 5808 . . . . 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 980 . . . 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 8820 . . . 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 2290 . . 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 2295 . 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 1158 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   CCcc 8703   1c1 8706    x. cmul 8710   2c2 9763   NN0cn0 9933   ZZcz 9992   ^cexp 11071   abscabs 11685    gcd cgcd 12648
This theorem is referenced by:  coprimeprodsq2  12826  pythagtriplem6  12837
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-divides 12495  df-gcd 12649
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