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Theorem lawcos 20660
Description: Law of Cosines. Given three distinct points A, B, and C, prove a relationship between their segment lengths. This theorem is expressed using the complex number plane as a plane, where  F is the signed angle construct (as used in ang180 20658),  X is the distance of line segment BC,  Y is the distance of line segment AC,  Z is the distance of line segment AB, and  O is the distinguished (signed) angle m/_ BCA on the complex plane. We translate triangle ABC to move C to the origin (C-C), B to U=(B-C), and A to V=(A-C), then use lemma lawcoslem1 20659 to prove this algebraically simpler case. The metamath convention is to use a signed angle; in this case the sign doesn't matter because we use the cosine of the angle (see cosneg 12750). The Pythagorean Theorem pythag 20661 is a special case of the law of cosines. The theorem's expression and approach were suggested by Mario Carneiro. (Contributed by David A. Wheeler, 12-Jun-2015.)
Hypotheses
Ref Expression
lawcos.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
lawcos.2  |-  X  =  ( abs `  ( B  -  C )
)
lawcos.3  |-  Y  =  ( abs `  ( A  -  C )
)
lawcos.4  |-  Z  =  ( abs `  ( A  -  B )
)
lawcos.5  |-  O  =  ( ( B  -  C ) F ( A  -  C ) )
Assertion
Ref Expression
lawcos  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^ 2 ) )  -  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    F( x, y)    O( x, y)    X( x, y)    Y( x, y)    Z( x, y)

Proof of Theorem lawcos
StepHypRef Expression
1 subcl 9307 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
213adant2 977 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
32adantr 453 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  e.  CC )
4 subcl 9307 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
543adant1 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
65adantr 453 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  e.  CC )
7 subeq0 9329 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
87necon3bid 2638 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =/=  0  <->  A  =/=  C ) )
98bicomd 194 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0 ) )
1093adant2 977 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0
) )
1110biimpa 472 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  A  =/=  C
)  ->  ( A  -  C )  =/=  0
)
1211adantrr 699 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  =/=  0 )
13 subeq0 9329 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
1413necon3bid 2638 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =/=  0  <->  B  =/=  C ) )
1514bicomd 194 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0 ) )
16153adant1 976 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0
) )
1716biimpa 472 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  B  =/=  C
)  ->  ( B  -  C )  =/=  0
)
1817adantrl 698 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  =/=  0 )
193, 6, 12, 18lawcoslem1 20659 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 )  =  ( ( ( ( abs `  ( A  -  C )
) ^ 2 )  +  ( ( abs `  ( B  -  C
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( A  -  C )
)  x.  ( abs `  ( B  -  C
) ) )  x.  ( ( Re `  ( ( A  -  C )  /  ( B  -  C )
) )  /  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) ) ) ) )
20 nnncan2 9340 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  ( B  -  C ) )  =  ( A  -  B ) )
2120fveq2d 5734 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( abs `  ( ( A  -  C )  -  ( B  -  C
) ) )  =  ( abs `  ( A  -  B )
) )
22 lawcos.4 . . . . 5  |-  Z  =  ( abs `  ( A  -  B )
)
2321, 22syl6reqr 2489 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  Z  =  ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) )
2423oveq1d 6098 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
2524adantr 453 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
263abscld 12240 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2726recnd 9116 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  CC )
2827sqcld 11523 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
) ^ 2 )  e.  CC )
296abscld 12240 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  RR )
3029recnd 9116 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  CC )
3130sqcld 11523 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( B  -  C )
) ^ 2 )  e.  CC )
3228, 31addcomd 9270 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( ( abs `  ( A  -  C )
) ^ 2 )  +  ( ( abs `  ( B  -  C
) ) ^ 2 ) )  =  ( ( ( abs `  ( B  -  C )
) ^ 2 )  +  ( ( abs `  ( A  -  C
) ) ^ 2 ) ) )
33 lawcos.2 . . . . . 6  |-  X  =  ( abs `  ( B  -  C )
)
3433oveq1i 6093 . . . . 5  |-  ( X ^ 2 )  =  ( ( abs `  ( B  -  C )
) ^ 2 )
35 lawcos.3 . . . . . 6  |-  Y  =  ( abs `  ( A  -  C )
)
3635oveq1i 6093 . . . . 5  |-  ( Y ^ 2 )  =  ( ( abs `  ( A  -  C )
) ^ 2 )
3734, 36oveq12i 6095 . . . 4  |-  ( ( X ^ 2 )  +  ( Y ^
2 ) )  =  ( ( ( abs `  ( B  -  C
) ) ^ 2 )  +  ( ( abs `  ( A  -  C ) ) ^ 2 ) )
3832, 37syl6reqr 2489 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( X ^ 2 )  +  ( Y ^ 2 ) )  =  ( ( ( abs `  ( A  -  C ) ) ^ 2 )  +  ( ( abs `  ( B  -  C )
) ^ 2 ) ) )
3927, 30mulcomd 9111 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
)  x.  ( abs `  ( B  -  C
) ) )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) ) )
4033, 35oveq12i 6095 . . . . . 6  |-  ( X  x.  Y )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) )
4139, 40syl6reqr 2489 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( X  x.  Y )  =  ( ( abs `  ( A  -  C
) )  x.  ( abs `  ( B  -  C ) ) ) )
42 lawcos.5 . . . . . . . . 9  |-  O  =  ( ( B  -  C ) F ( A  -  C ) )
4342fveq2i 5733 . . . . . . . 8  |-  ( cos `  O )  =  ( cos `  ( ( B  -  C ) F ( A  -  C ) ) )
44 lawcos.1 . . . . . . . . . 10  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
4544, 6, 18, 3, 12angvald 20648 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( B  -  C
) F ( A  -  C ) )  =  ( Im `  ( log `  ( ( A  -  C )  /  ( B  -  C ) ) ) ) )
4645fveq2d 5734 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  ( ( B  -  C ) F ( A  -  C
) ) )  =  ( cos `  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) )
4743, 46syl5eq 2482 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  O )  =  ( cos `  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) )
483, 6, 18divcld 9792 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  e.  CC )
493, 6, 12, 18divne0d 9808 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  =/=  0 )
5048, 49logcld 20470 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( log `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  CC )
5150imcld 12002 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR )
52 recosval 12739 . . . . . . . 8  |-  ( ( Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR  ->  ( cos `  ( Im `  ( log `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) )  =  ( Re
`  ( exp `  (
_i  x.  ( Im `  ( log `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) ) ) ) )
5351, 52syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  ( Im `  ( log `  ( ( A  -  C )  /  ( B  -  C ) ) ) ) )  =  ( Re `  ( exp `  ( _i  x.  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) ) ) )
5447, 53eqtrd 2470 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  O )  =  ( Re `  ( exp `  ( _i  x.  ( Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) ) ) )
55 efiarg 20504 . . . . . . . 8  |-  ( ( ( ( A  -  C )  /  ( B  -  C )
)  e.  CC  /\  ( ( A  -  C )  /  ( B  -  C )
)  =/=  0 )  ->  ( exp `  (
_i  x.  ( Im `  ( log `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) ) )  =  ( ( ( A  -  C )  /  ( B  -  C ) )  / 
( abs `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) )
5648, 49, 55syl2anc 644 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( exp `  ( _i  x.  ( Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) )  =  ( ( ( A  -  C
)  /  ( B  -  C ) )  /  ( abs `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) )
5756fveq2d 5734 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Re `  ( exp `  ( _i  x.  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) ) )  =  ( Re `  ( ( ( A  -  C
)  /  ( B  -  C ) )  /  ( abs `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) ) )
5848abscld 12240 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  RR )
5948, 49absne0d 12251 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  =/=  0 )
6058, 48, 59redivd 12036 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Re `  ( (
( A  -  C
)  /  ( B  -  C ) )  /  ( abs `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) )  =  ( ( Re `  ( ( A  -  C )  /  ( B  -  C )
) )  /  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) )
6154, 57, 603eqtrd 2474 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  O )  =  ( ( Re `  ( ( A  -  C )  /  ( B  -  C )
) )  /  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) )
6241, 61oveq12d 6101 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( X  x.  Y
)  x.  ( cos `  O ) )  =  ( ( ( abs `  ( A  -  C
) )  x.  ( abs `  ( B  -  C ) ) )  x.  ( ( Re
`  ( ( A  -  C )  / 
( B  -  C
) ) )  / 
( abs `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) ) )
6362oveq2d 6099 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) )  =  ( 2  x.  (
( ( abs `  ( A  -  C )
)  x.  ( abs `  ( B  -  C
) ) )  x.  ( ( Re `  ( ( A  -  C )  /  ( B  -  C )
) )  /  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) ) ) )
6438, 63oveq12d 6101 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( ( X ^
2 )  +  ( Y ^ 2 ) )  -  ( 2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) )  =  ( ( ( ( abs `  ( A  -  C )
) ^ 2 )  +  ( ( abs `  ( B  -  C
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( A  -  C )
)  x.  ( abs `  ( B  -  C
) ) )  x.  ( ( Re `  ( ( A  -  C )  /  ( B  -  C )
) )  /  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) ) ) ) ) ) )
6519, 25, 643eqtr4d 2480 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( ( X ^ 2 )  +  ( Y ^ 2 ) )  -  (
2  x.  ( ( X  x.  Y )  x.  ( cos `  O
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   CCcc 8990   RRcr 8991   0cc0 8992   _ici 8994    + caddc 8995    x. cmul 8997    - cmin 9293    / cdiv 9679   2c2 10051   ^cexp 11384   Recre 11904   Imcim 11905   abscabs 12041   expce 12666   cosccos 12669   logclog 20454
This theorem is referenced by:  pythag  20661  ssscongptld  20668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482  df-ef 12672  df-sin 12674  df-cos 12675  df-pi 12677  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456
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