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Theorem lawcos 20116
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 20114),  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 20115 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 12429). The Pythagorean Theorem pythag 20117 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 9053 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  -  C
)  e.  CC )
213adant2 974 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  C )  e.  CC )
32adantr 451 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  e.  CC )
4 subcl 9053 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
543adant1 973 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
65adantr 451 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  e.  CC )
7 subeq0 9075 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
87necon3bid 2483 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =/=  0  <->  A  =/=  C ) )
98bicomd 192 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0 ) )
1093adant2 974 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  =/=  C  <->  ( A  -  C )  =/=  0
) )
1110biimpa 470 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  A  =/=  C
)  ->  ( A  -  C )  =/=  0
)
1211adantrr 697 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( A  -  C )  =/=  0 )
13 subeq0 9075 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
1413necon3bid 2483 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =/=  0  <->  B  =/=  C ) )
1514bicomd 192 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0 ) )
16153adant1 973 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  =/=  C  <->  ( B  -  C )  =/=  0
) )
1716biimpa 470 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  B  =/=  C
)  ->  ( B  -  C )  =/=  0
)
1817adantrl 696 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( B  -  C )  =/=  0 )
193, 6, 12, 18lawcoslem1 20115 . 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 9086 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  ( B  -  C ) )  =  ( A  -  B ) )
2120fveq2d 5531 . . . . 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 2336 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  Z  =  ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) )
2423oveq1d 5875 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
2524adantr 451 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( Z ^ 2 )  =  ( ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) ^ 2 ) )
263abscld 11920 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2726recnd 8863 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  CC )
2827sqcld 11245 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
) ^ 2 )  e.  CC )
296abscld 11920 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  RR )
3029recnd 8863 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  CC )
3130sqcld 11245 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( B  -  C )
) ^ 2 )  e.  CC )
3228, 31addcomd 9016 . . . 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 5870 . . . . 5  |-  ( X ^ 2 )  =  ( ( abs `  ( B  -  C )
) ^ 2 )
35 lawcos.3 . . . . . 6  |-  Y  =  ( abs `  ( A  -  C )
)
3635oveq1i 5870 . . . . 5  |-  ( Y ^ 2 )  =  ( ( abs `  ( A  -  C )
) ^ 2 )
3734, 36oveq12i 5872 . . . 4  |-  ( ( X ^ 2 )  +  ( Y ^
2 ) )  =  ( ( ( abs `  ( B  -  C
) ) ^ 2 )  +  ( ( abs `  ( A  -  C ) ) ^ 2 ) )
3832, 37syl6reqr 2336 . . 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 8858 . . . . . 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 5872 . . . . . 6  |-  ( X  x.  Y )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) )
4139, 40syl6reqr 2336 . . . . 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 5530 . . . . . . . 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 20101 . . . . . . . . . 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 1183 . . . . . . . . 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 5531 . . . . . . . 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 2329 . . . . . . 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 9538 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  e.  CC )
503, 6, 12, 18divne0d 9554 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  =/=  0 )
51 logcl 19928 . . . . . . . . . 10  |-  ( ( ( ( A  -  C )  /  ( B  -  C )
)  e.  CC  /\  ( ( A  -  C )  /  ( B  -  C )
)  =/=  0 )  ->  ( log `  (
( A  -  C
)  /  ( B  -  C ) ) )  e.  CC )
5249, 50, 51syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( log `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  CC )
5352imcld 11682 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR )
54 recosval 12418 . . . . . . . 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 15 . . . . . . 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 2317 . . . . . 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 19963 . . . . . . . 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 642 . . . . . . 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 5531 . . . . . 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 11920 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  RR )
6149, 50absne0d 11931 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  =/=  0 )
6260, 49, 61redivd 11716 . . . . . 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 2321 . . . . 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 5878 . . . 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 5876 . . 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 5878 . 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 2327 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448    \ cdif 3151   {csn 3642   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   CCcc 8737   RRcr 8738   0cc0 8739   _ici 8741    + caddc 8742    x. cmul 8744    - cmin 9039    / cdiv 9425   2c2 9797   ^cexp 11106   Recre 11584   Imcim 11585   abscabs 11721   expce 12345   cosccos 12348   logclog 19914
This theorem is referenced by:  pythag  20117  ssscongptld  20124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916
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