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Theorem pythagtriplem11 13199
Description: Lemma for pythagtrip 13208. Show that  M (which will eventually be closely related to the  m in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypothesis
Ref Expression
pythagtriplem11.1  |-  M  =  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 )
Assertion
Ref Expression
pythagtriplem11  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  M  e.  NN )

Proof of Theorem pythagtriplem11
StepHypRef Expression
1 pythagtriplem11.1 . 2  |-  M  =  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 )
2 pythagtriplem9 13198 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  +  B ) )  e.  NN )
32nnzd 10374 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  +  B ) )  e.  ZZ )
4 simp3r 986 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  A )
5 nnz 10303 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
653ad2ant3 980 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  ZZ )
7 nnz 10303 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
873ad2ant2 979 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  ZZ )
96, 8zaddcld 10379 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  ZZ )
1093ad2ant1 978 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  +  B )  e.  ZZ )
11 nnz 10303 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
12113ad2ant1 978 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
13123ad2ant1 978 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  A  e.  ZZ )
14 2z 10312 . . . . . . . . . . 11  |-  2  e.  ZZ
15 dvdsgcdb 13044 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  +  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  +  B )  gcd  A
) ) )
1614, 15mp3an1 1266 . . . . . . . . . 10  |-  ( ( ( C  +  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  +  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  +  B )  gcd  A
) ) )
1710, 13, 16syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( 2  ||  ( C  +  B )  /\  2  ||  A )  <->  2  ||  ( ( C  +  B )  gcd  A ) ) )
1817biimpar 472 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  2  ||  ( ( C  +  B )  gcd  A
) )  ->  (
2  ||  ( C  +  B )  /\  2  ||  A ) )
1918simprd 450 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  2  ||  ( ( C  +  B )  gcd  A
) )  ->  2  ||  A )
204, 19mtand 641 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  ( ( C  +  B )  gcd 
A ) )
21 pythagtriplem7 13196 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  +  B ) )  =  ( ( C  +  B )  gcd  A
) )
2221breq2d 4224 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  ||  ( sqr `  ( C  +  B
) )  <->  2  ||  ( ( C  +  B )  gcd  A
) ) )
2320, 22mtbird 293 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  ( sqr `  ( C  +  B )
) )
24 pythagtriplem8 13197 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  e.  NN )
2524nnzd 10374 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  e.  ZZ )
266, 8zsubcld 10380 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
27263ad2ant1 978 . . . . . . . . . 10  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( C  -  B )  e.  ZZ )
28 dvdsgcdb 13044 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  ( C  -  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  -  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  -  B )  gcd  A
) ) )
2914, 28mp3an1 1266 . . . . . . . . . 10  |-  ( ( ( C  -  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  -  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  -  B )  gcd  A
) ) )
3027, 13, 29syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( 2  ||  ( C  -  B )  /\  2  ||  A )  <->  2  ||  ( ( C  -  B )  gcd  A ) ) )
3130biimpar 472 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  2  ||  ( ( C  -  B )  gcd  A
) )  ->  (
2  ||  ( C  -  B )  /\  2  ||  A ) )
3231simprd 450 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  (
( A ^ 2 )  +  ( B ^ 2 ) )  =  ( C ^
2 )  /\  (
( A  gcd  B
)  =  1  /\ 
-.  2  ||  A
) )  /\  2  ||  ( ( C  -  B )  gcd  A
) )  ->  2  ||  A )
334, 32mtand 641 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  ( ( C  -  B )  gcd 
A ) )
34 pythagtriplem6 13195 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  =  ( ( C  -  B )  gcd  A
) )
3534breq2d 4224 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  ||  ( sqr `  ( C  -  B
) )  <->  2  ||  ( ( C  -  B )  gcd  A
) ) )
3633, 35mtbird 293 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  -.  2  ||  ( sqr `  ( C  -  B )
) )
37 opoe 13185 . . . . 5  |-  ( ( ( ( sqr `  ( C  +  B )
)  e.  ZZ  /\  -.  2  ||  ( sqr `  ( C  +  B
) ) )  /\  ( ( sqr `  ( C  -  B )
)  e.  ZZ  /\  -.  2  ||  ( sqr `  ( C  -  B
) ) ) )  ->  2  ||  (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) ) )
383, 23, 25, 36, 37syl22anc 1185 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  2  ||  ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) ) )
392, 24nnaddcld 10046 . . . . . 6  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  NN )
4039nnzd 10374 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  ZZ )
41 2ne0 10083 . . . . . 6  |-  2  =/=  0
42 dvdsval2 12855 . . . . . 6  |-  ( ( 2  e.  ZZ  /\  2  =/=  0  /\  (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  ZZ )  ->  (
2  ||  ( ( sqr `  ( C  +  B ) )  +  ( sqr `  ( C  -  B )
) )  <->  ( (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 )  e.  ZZ ) )
4314, 41, 42mp3an12 1269 . . . . 5  |-  ( ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  ZZ  ->  ( 2 
||  ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  <->  ( ( ( sqr `  ( C  +  B ) )  +  ( sqr `  ( C  -  B )
) )  /  2
)  e.  ZZ ) )
4440, 43syl 16 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
2  ||  ( ( sqr `  ( C  +  B ) )  +  ( sqr `  ( C  -  B )
) )  <->  ( (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 )  e.  ZZ ) )
4538, 44mpbid 202 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 )  e.  ZZ )
462nnred 10015 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  +  B ) )  e.  RR )
4724nnred 10015 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  ( sqr `  ( C  -  B ) )  e.  RR )
482nngt0d 10043 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <  ( sqr `  ( C  +  B )
) )
4924nngt0d 10043 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <  ( sqr `  ( C  -  B )
) )
5046, 47, 48, 49addgt0d 9601 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <  ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) ) )
5139nnred 10015 . . . . 5  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  RR )
52 halfpos2 10197 . . . . 5  |-  ( ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  e.  RR  ->  ( 0  <  ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  <->  0  <  (
( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 ) ) )
5351, 52syl 16 . . . 4  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
0  <  ( ( sqr `  ( C  +  B ) )  +  ( sqr `  ( C  -  B )
) )  <->  0  <  ( ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 ) ) )
5450, 53mpbid 202 . . 3  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  0  <  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 ) )
55 elnnz 10292 . . 3  |-  ( ( ( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 )  e.  NN  <->  ( ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 )  e.  ZZ  /\  0  <  ( ( ( sqr `  ( C  +  B
) )  +  ( sqr `  ( C  -  B ) ) )  /  2 ) ) )
5645, 54, 55sylanbrc 646 . 2  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  (
( ( sqr `  ( C  +  B )
)  +  ( sqr `  ( C  -  B
) ) )  / 
2 )  e.  NN )
571, 56syl5eqel 2520 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  /\  ( ( A ^
2 )  +  ( B ^ 2 ) )  =  ( C ^ 2 )  /\  ( ( A  gcd  B )  =  1  /\ 
-.  2  ||  A
) )  ->  M  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   ZZcz 10282   ^cexp 11382   sqrcsqr 12038    || cdivides 12852    gcd cgcd 13006
This theorem is referenced by:  pythagtriplem18  13206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080
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