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Theorem pythagtriplem11 12925
Description: Lemma for pythagtrip 12934. 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 12924 . . . . . 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 10163 . . . . 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 984 . . . . . . 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 10092 . . . . . . . . . . . . 13  |-  ( C  e.  NN  ->  C  e.  ZZ )
653ad2ant3 978 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  C  e.  ZZ )
7 nnz 10092 . . . . . . . . . . . . 13  |-  ( B  e.  NN  ->  B  e.  ZZ )
873ad2ant2 977 . . . . . . . . . . . 12  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  B  e.  ZZ )
96, 8zaddcld 10168 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  +  B )  e.  ZZ )
1093ad2ant1 976 . . . . . . . . . 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 10092 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  e.  ZZ )
12113ad2ant1 976 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  A  e.  ZZ )
13123ad2ant1 976 . . . . . . . . . 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 10101 . . . . . . . . . . 11  |-  2  e.  ZZ
15 dvdsgcdb 12770 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  ( C  +  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  +  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  +  B )  gcd  A
) ) )
1614, 15mp3an1 1264 . . . . . . . . . 10  |-  ( ( ( C  +  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  +  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  +  B )  gcd  A
) ) )
1710, 13, 16syl2anc 642 . . . . . . . . 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 471 . . . . . . . 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 449 . . . . . . 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 640 . . . . . 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 12922 . . . . . . 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 4072 . . . . . 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 292 . . . . 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 12923 . . . . . 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 10163 . . . . 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 10169 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  NN )  ->  ( C  -  B )  e.  ZZ )
27263ad2ant1 976 . . . . . . . . . 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 12770 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  ( C  -  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  -  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  -  B )  gcd  A
) ) )
2914, 28mp3an1 1264 . . . . . . . . . 10  |-  ( ( ( C  -  B
)  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( 2  ||  ( C  -  B
)  /\  2  ||  A )  <->  2  ||  ( ( C  -  B )  gcd  A
) ) )
3027, 13, 29syl2anc 642 . . . . . . . . 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 471 . . . . . . . 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 449 . . . . . . 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 640 . . . . . 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 12921 . . . . . . 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 4072 . . . . . 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 292 . . . . 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 12911 . . . . 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 1183 . . . 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 9837 . . . . . 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 10163 . . . . 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 9874 . . . . . 6  |-  2  =/=  0
42 dvdsval2 12581 . . . . . 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 1267 . . . . 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 15 . . . 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 201 . . 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 9806 . . . . 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 9806 . . . . 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 9834 . . . . 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 9834 . . . . 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 9392 . . . 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 9806 . . . . 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 9988 . . . . 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 15 . . . 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 201 . . 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 10081 . . 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 645 . 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 2400 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    < clt 8912    - cmin 9082    / cdiv 9468   NNcn 9791   2c2 9840   ZZcz 10071   ^cexp 11151   sqrcsqr 11765    || cdivides 12578    gcd cgcd 12732
This theorem is referenced by:  pythagtriplem18  12932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-dvds 12579  df-gcd 12733  df-prm 12806
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