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Theorem lawcos 20077
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 20075),  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 20076 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 12390). The Pythagorean Theorem pythag 20078 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 9019 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
213adant2 979 . . . 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 9019 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
543adant1 978 . . . 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 9041 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
87necon3bid 2456 . . . . . . 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 979 . . . . 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 700 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  =/=  0 )
13 subeq0 9041 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
1413necon3bid 2456 . . . . . . 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 978 . . . . 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 699 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  =/=  0 )
193, 6, 12, 18lawcoslem1 20076 . 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 9052 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  ( B  -  C ) )  =  ( A  -  B ) )
2120fveq2d 5462 . . . . 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 2309 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  Z  =  ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) )
2423oveq1d 5807 . . 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 11884 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2726recnd 8829 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  CC )
2827sqcld 11210 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
) ^ 2 )  e.  CC )
296abscld 11884 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  RR )
3029recnd 8829 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  CC )
3130sqcld 11210 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( B  -  C )
) ^ 2 )  e.  CC )
3228, 31addcomd 8982 . . . 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 5802 . . . . 5  |-  ( X ^ 2 )  =  ( ( abs `  ( B  -  C )
) ^ 2 )
35 lawcos.3 . . . . . 6  |-  Y  =  ( abs `  ( A  -  C )
)
3635oveq1i 5802 . . . . 5  |-  ( Y ^ 2 )  =  ( ( abs `  ( A  -  C )
) ^ 2 )
3734, 36oveq12i 5804 . . . 4  |-  ( ( X ^ 2 )  +  ( Y ^
2 ) )  =  ( ( ( abs `  ( B  -  C
) ) ^ 2 )  +  ( ( abs `  ( A  -  C ) ) ^ 2 ) )
3832, 37syl6reqr 2309 . . 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 8824 . . . . . 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 5804 . . . . . 6  |-  ( X  x.  Y )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) )
4139, 40syl6reqr 2309 . . . . 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 5461 . . . . . . . 8  |-  ( cos `  O )  =  ( cos `  ( ( B  -  C ) F ( A  -  C ) ) )
44 lawcos.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
4544angval 20062 . . . . . . . . . 10  |-  ( ( ( ( B  -  C )  e.  CC  /\  ( B  -  C
)  =/=  0 )  /\  ( ( A  -  C )  e.  CC  /\  ( A  -  C )  =/=  0 ) )  -> 
( ( B  -  C ) F ( A  -  C ) )  =  ( Im
`  ( log `  (
( A  -  C
)  /  ( B  -  C ) ) ) ) )
466, 18, 3, 12, 45syl22anc 1188 . . . . . . . . 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 ) ) ) ) )
4746fveq2d 5462 . . . . . . . 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 )
) ) ) ) )
4843, 47syl5eq 2302 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( cos `  O )  =  ( cos `  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) ) ) )
493, 6, 18divcld 9504 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  e.  CC )
503, 6, 12, 18divne0d 9520 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  =/=  0 )
51 logcl 19889 . . . . . . . . . 10  |-  ( ( ( ( A  -  C )  /  ( B  -  C )
)  e.  CC  /\  ( ( A  -  C )  /  ( B  -  C )
)  =/=  0 )  ->  ( log `  (
( A  -  C
)  /  ( B  -  C ) ) )  e.  CC )
5249, 50, 51syl2anc 645 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( log `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  CC )
5352imcld 11646 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR )
54 recosval 12379 . . . . . . . 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 ) ) ) ) ) ) ) )
5553, 54syl 17 . . . . . . 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 )
) ) ) ) ) ) )
5648, 55eqtrd 2290 . . . . . 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 )
) ) ) ) ) ) )
57 efiarg 19924 . . . . . . . 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 ) ) ) ) )
5849, 50, 57syl2anc 645 . . . . . . 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 ) ) ) ) )
5958fveq2d 5462 . . . . . 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 ) ) ) ) ) )
6049abscld 11884 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  RR )
6149, 50absne0d 11895 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  =/=  0 )
6260, 49, 61redivd 11680 . . . . . 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
) ) ) ) )
6356, 59, 623eqtrd 2294 . . . . 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
) ) ) ) )
6441, 63oveq12d 5810 . . . 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 ) ) ) ) ) )
6564oveq2d 5808 . . 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
) ) ) ) ) ) )
6638, 65oveq12d 5810 . 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
) ) ) ) ) ) ) )
6719, 25, 663eqtr4d 2300 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421    \ cdif 3124   {csn 3614   ` cfv 4673  (class class class)co 5792    e. cmpt2 5794   CCcc 8703   RRcr 8704   0cc0 8705   _ici 8707    + caddc 8708    x. cmul 8710    - cmin 9005    / cdiv 9391   2c2 9763   ^cexp 11071   Recre 11548   Imcim 11549   abscabs 11685   expce 12306   cosccos 12309   logclog 19875
This theorem is referenced by:  pythag  20078  ssscongptld  20085
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-shft 11528  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-limsup 11911  df-clim 11928  df-rlim 11929  df-sum 12125  df-ef 12312  df-sin 12314  df-cos 12315  df-pi 12317  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-limc 19179  df-dv 19180  df-log 19877
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