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Theorem lawcos 20611
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 20609),  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 20610 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 12703). The Pythagorean Theorem pythag 20612 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 9261 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
213adant2 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
32adantr 452 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  e.  CC )
4 subcl 9261 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
543adant1 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
65adantr 452 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  e.  CC )
7 subeq0 9283 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
87necon3bid 2602 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =/=  0  <->  A  =/=  C ) )
98bicomd 193 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0 ) )
1093adant2 976 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0
) )
1110biimpa 471 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  A  =/=  C
)  ->  ( A  -  C )  =/=  0
)
1211adantrr 698 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  =/=  0 )
13 subeq0 9283 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
1413necon3bid 2602 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =/=  0  <->  B  =/=  C ) )
1514bicomd 193 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0 ) )
16153adant1 975 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0
) )
1716biimpa 471 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  B  =/=  C
)  ->  ( B  -  C )  =/=  0
)
1817adantrl 697 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  =/=  0 )
193, 6, 12, 18lawcoslem1 20610 . 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 9294 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  ( B  -  C ) )  =  ( A  -  B ) )
2120fveq2d 5691 . . . . 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 2455 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  Z  =  ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) )
2423oveq1d 6055 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
2524adantr 452 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
263abscld 12193 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2726recnd 9070 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  CC )
2827sqcld 11476 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
) ^ 2 )  e.  CC )
296abscld 12193 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  RR )
3029recnd 9070 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  CC )
3130sqcld 11476 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( B  -  C )
) ^ 2 )  e.  CC )
3228, 31addcomd 9224 . . . 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 6050 . . . . 5  |-  ( X ^ 2 )  =  ( ( abs `  ( B  -  C )
) ^ 2 )
35 lawcos.3 . . . . . 6  |-  Y  =  ( abs `  ( A  -  C )
)
3635oveq1i 6050 . . . . 5  |-  ( Y ^ 2 )  =  ( ( abs `  ( A  -  C )
) ^ 2 )
3734, 36oveq12i 6052 . . . 4  |-  ( ( X ^ 2 )  +  ( Y ^
2 ) )  =  ( ( ( abs `  ( B  -  C
) ) ^ 2 )  +  ( ( abs `  ( A  -  C ) ) ^ 2 ) )
3832, 37syl6reqr 2455 . . 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 9065 . . . . . 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 6052 . . . . . 6  |-  ( X  x.  Y )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) )
4139, 40syl6reqr 2455 . . . . 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 5690 . . . . . . . 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 20599 . . . . . . . . 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 5691 . . . . . . . 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 2448 . . . . . . 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 9746 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  e.  CC )
493, 6, 12, 18divne0d 9762 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  =/=  0 )
5048, 49logcld 20421 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( log `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  CC )
5150imcld 11955 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR )
52 recosval 12692 . . . . . . . 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 2436 . . . . . 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 20455 . . . . . . . 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 643 . . . . . . 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 5691 . . . . . 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 12193 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  RR )
5948, 49absne0d 12204 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  =/=  0 )
6058, 48, 59redivd 11989 . . . . . 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 2440 . . . . 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 6058 . . . 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 6056 . . 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 6058 . 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 2446 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   _ici 8948    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633   2c2 10005   ^cexp 11337   Recre 11857   Imcim 11858   abscabs 11994   expce 12619   cosccos 12622   logclog 20405
This theorem is referenced by:  pythag  20612  ssscongptld  20619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407
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