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Theorem ang180 20217
Description: The sum of angles  m A B C  +  m B C A  +  m C A B in a triangle adds up to either  pi or  -u pi, i.e. 180 degrees. (The sign is due to the two possible orientations of vertex arrangement and our signed notion of angle). (Contributed by Mario Carneiro, 23-Sep-2014.)
Hypothesis
Ref Expression
ang.1  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
Assertion
Ref Expression
ang180  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( ( ( C  -  B ) F ( A  -  B
) )  +  ( ( A  -  C
) F ( B  -  C ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e.  { -u pi ,  pi } )
Distinct variable groups:    x, y, A    x, B, y    x, C, y
Allowed substitution hints:    F( x, y)

Proof of Theorem ang180
StepHypRef Expression
1 simpl3 960 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  C  e.  CC )
2 simpl2 959 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  B  e.  CC )
31, 2subcld 9244 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( C  -  B )  e.  CC )
4 simpr2 962 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  B  =/=  C )
54necomd 2604 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  C  =/=  B )
6 subeq0 9160 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( ( C  -  B )  =  0  <-> 
C  =  B ) )
71, 2, 6syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  B
)  =  0  <->  C  =  B ) )
87necon3bid 2556 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  B
)  =/=  0  <->  C  =/=  B ) )
95, 8mpbird 223 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( C  -  B )  =/=  0 )
10 simpl1 958 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  A  e.  CC )
1110, 2subcld 9244 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( A  -  B )  e.  CC )
12 simpr1 961 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  A  =/=  B )
13 subeq0 9160 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <-> 
A  =  B ) )
1410, 2, 13syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( A  -  B
)  =  0  <->  A  =  B ) )
1514necon3bid 2556 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( A  -  B
)  =/=  0  <->  A  =/=  B ) )
1612, 15mpbird 223 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( A  -  B )  =/=  0 )
17 ang.1 . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
1817angneg 20206 . . . . . 6  |-  ( ( ( ( C  -  B )  e.  CC  /\  ( C  -  B
)  =/=  0 )  /\  ( ( A  -  B )  e.  CC  /\  ( A  -  B )  =/=  0 ) )  -> 
( -u ( C  -  B ) F -u ( A  -  B
) )  =  ( ( C  -  B
) F ( A  -  B ) ) )
193, 9, 11, 16, 18syl22anc 1183 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( -u ( C  -  B
) F -u ( A  -  B )
)  =  ( ( C  -  B ) F ( A  -  B ) ) )
201, 2negsubdi2d 9260 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( C  -  B )  =  ( B  -  C ) )
212, 1, 10nnncan2d 9279 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  -  ( C  -  A ) )  =  ( B  -  C ) )
2220, 21eqtr4d 2393 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( C  -  B )  =  ( ( B  -  A )  -  ( C  -  A
) ) )
2310, 2negsubdi2d 9260 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
2422, 23oveq12d 5960 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( -u ( C  -  B
) F -u ( A  -  B )
)  =  ( ( ( B  -  A
)  -  ( C  -  A ) ) F ( B  -  A ) ) )
2519, 24eqtr3d 2392 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  B
) F ( A  -  B ) )  =  ( ( ( B  -  A )  -  ( C  -  A ) ) F ( B  -  A
) ) )
2610, 1subcld 9244 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( A  -  C )  e.  CC )
27 simpr3 963 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  A  =/=  C )
28 subeq0 9160 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
2910, 1, 28syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( A  -  C
)  =  0  <->  A  =  C ) )
3029necon3bid 2556 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( A  -  C
)  =/=  0  <->  A  =/=  C ) )
3127, 30mpbird 223 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( A  -  C )  =/=  0 )
322, 1subcld 9244 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  C )  e.  CC )
33 subeq0 9160 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
342, 1, 33syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  C
)  =  0  <->  B  =  C ) )
3534necon3bid 2556 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  C
)  =/=  0  <->  B  =/=  C ) )
364, 35mpbird 223 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  C )  =/=  0 )
3717angneg 20206 . . . . . 6  |-  ( ( ( ( A  -  C )  e.  CC  /\  ( A  -  C
)  =/=  0 )  /\  ( ( B  -  C )  e.  CC  /\  ( B  -  C )  =/=  0 ) )  -> 
( -u ( A  -  C ) F -u ( B  -  C
) )  =  ( ( A  -  C
) F ( B  -  C ) ) )
3826, 31, 32, 36, 37syl22anc 1183 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( -u ( A  -  C
) F -u ( B  -  C )
)  =  ( ( A  -  C ) F ( B  -  C ) ) )
3910, 1negsubdi2d 9260 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( A  -  C )  =  ( C  -  A ) )
402, 1negsubdi2d 9260 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( B  -  C )  =  ( C  -  B ) )
411, 2, 10nnncan2d 9279 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  A
)  -  ( B  -  A ) )  =  ( C  -  B ) )
4240, 41eqtr4d 2393 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( B  -  C )  =  ( ( C  -  A )  -  ( B  -  A
) ) )
4339, 42oveq12d 5960 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( -u ( A  -  C
) F -u ( B  -  C )
)  =  ( ( C  -  A ) F ( ( C  -  A )  -  ( B  -  A
) ) ) )
4438, 43eqtr3d 2392 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( A  -  C
) F ( B  -  C ) )  =  ( ( C  -  A ) F ( ( C  -  A )  -  ( B  -  A )
) ) )
4525, 44oveq12d 5960 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( ( C  -  B ) F ( A  -  B ) )  +  ( ( A  -  C ) F ( B  -  C ) ) )  =  ( ( ( ( B  -  A
)  -  ( C  -  A ) ) F ( B  -  A ) )  +  ( ( C  -  A ) F ( ( C  -  A
)  -  ( B  -  A ) ) ) ) )
4645oveq1d 5957 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( ( ( C  -  B ) F ( A  -  B
) )  +  ( ( A  -  C
) F ( B  -  C ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  =  ( ( ( ( ( B  -  A )  -  ( C  -  A )
) F ( B  -  A ) )  +  ( ( C  -  A ) F ( ( C  -  A )  -  ( B  -  A )
) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) ) )
472, 10subcld 9244 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  A )  e.  CC )
4812necomd 2604 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  B  =/=  A )
49 subeq0 9160 . . . . . 6  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( B  -  A )  =  0  <-> 
B  =  A ) )
502, 10, 49syl2anc 642 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  =  0  <->  B  =  A ) )
5150necon3bid 2556 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  =/=  0  <->  B  =/=  A ) )
5248, 51mpbird 223 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  A )  =/=  0 )
531, 10subcld 9244 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( C  -  A )  e.  CC )
5427necomd 2604 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  C  =/=  A )
55 subeq0 9160 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( ( C  -  A )  =  0  <-> 
C  =  A ) )
561, 10, 55syl2anc 642 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  A
)  =  0  <->  C  =  A ) )
5756necon3bid 2556 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  A
)  =/=  0  <->  C  =/=  A ) )
5854, 57mpbird 223 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( C  -  A )  =/=  0 )
59 subcan2 9159 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  A  e.  CC )  ->  (
( B  -  A
)  =  ( C  -  A )  <->  B  =  C ) )
602, 1, 10, 59syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  =  ( C  -  A )  <->  B  =  C ) )
6160necon3bid 2556 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  =/=  ( C  -  A )  <->  B  =/=  C ) )
624, 61mpbird 223 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  A )  =/=  ( C  -  A
) )
6317ang180lem5 20216 . . 3  |-  ( ( ( ( B  -  A )  e.  CC  /\  ( B  -  A
)  =/=  0 )  /\  ( ( C  -  A )  e.  CC  /\  ( C  -  A )  =/=  0 )  /\  ( B  -  A )  =/=  ( C  -  A
) )  ->  (
( ( ( ( B  -  A )  -  ( C  -  A ) ) F ( B  -  A
) )  +  ( ( C  -  A
) F ( ( C  -  A )  -  ( B  -  A ) ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e.  { -u pi ,  pi } )
6447, 52, 53, 58, 62, 63syl221anc 1193 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( ( ( ( B  -  A )  -  ( C  -  A ) ) F ( B  -  A
) )  +  ( ( C  -  A
) F ( ( C  -  A )  -  ( B  -  A ) ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e.  { -u pi ,  pi } )
6546, 64eqeltrd 2432 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( ( ( C  -  B ) F ( A  -  B
) )  +  ( ( A  -  C
) F ( B  -  C ) ) )  +  ( ( B  -  A ) F ( C  -  A ) ) )  e.  { -u pi ,  pi } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225   {csn 3716   {cpr 3717   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944   CCcc 8822   0cc0 8824    + caddc 8827    - cmin 9124   -ucneg 9125    / cdiv 9510   Imcim 11673   picpi 12439   logclog 20013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ioc 10750  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-fac 11379  df-bc 11406  df-hash 11428  df-shft 11652  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-limsup 12035  df-clim 12052  df-rlim 12053  df-sum 12250  df-ef 12440  df-sin 12442  df-cos 12443  df-pi 12445  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cncf 18479  df-limc 19314  df-dv 19315  df-log 20015
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