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Theorem ang180 20112
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 9157 . . . . . 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 2529 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  C  =/=  B )
6 subeq0 9073 . . . . . . . . 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 2481 . . . . . . 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 9157 . . . . . 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 9073 . . . . . . . . 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 2481 . . . . . . 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 20101 . . . . . 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 9173 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( C  -  B )  =  ( B  -  C ) )
212, 1, 10nnncan2d 9192 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( B  -  A
)  -  ( C  -  A ) )  =  ( B  -  C ) )
2220, 21eqtr4d 2318 . . . . . 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 9173 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( A  -  B )  =  ( B  -  A ) )
2422, 23oveq12d 5876 . . . . 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 2317 . . . 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 9157 . . . . . 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 9073 . . . . . . . . 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 2481 . . . . . . 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 9157 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  C )  e.  CC )
33 subeq0 9073 . . . . . . . . 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 2481 . . . . . . 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 20101 . . . . . 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 9173 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( A  -  C )  =  ( C  -  A ) )
402, 1negsubdi2d 9173 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  -u ( B  -  C )  =  ( C  -  B ) )
411, 2, 10nnncan2d 9192 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  (
( C  -  A
)  -  ( B  -  A ) )  =  ( C  -  B ) )
4240, 41eqtr4d 2318 . . . . . 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 5876 . . . . 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 2317 . . . 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 5876 . . 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 5873 . 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 9157 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( B  -  A )  e.  CC )
4812necomd 2529 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  B  =/=  A )
49 subeq0 9073 . . . . . 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 2481 . . . 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 9157 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  ( C  -  A )  e.  CC )
5427necomd 2529 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  B  /\  B  =/=  C  /\  A  =/=  C
) )  ->  C  =/=  A )
55 subeq0 9073 . . . . . 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 2481 . . . 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 9072 . . . . . 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 2481 . . . 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 20111 . . 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 2357 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   {cpr 3641   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   0cc0 8737    + caddc 8740    - cmin 9037   -ucneg 9038    / cdiv 9423   Imcim 11583   picpi 12348   logclog 19912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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