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Theorem lawcos 20109
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 20107),  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 20108 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 12422). The Pythagorean Theorem pythag 20110 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 9046 . . . . 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 9046 . . . . 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 9068 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  0  <-> 
A  =  C ) )
87necon3bid 2481 . . . . . . 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 9068 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( B  -  C )  =  0  <-> 
B  =  C ) )
1413necon3bid 2481 . . . . . . 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 20108 . 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 9079 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  C
)  -  ( B  -  C ) )  =  ( A  -  B ) )
2120fveq2d 5489 . . . . 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 2334 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  Z  =  ( abs `  (
( A  -  C
)  -  ( B  -  C ) ) ) )
2423oveq1d 5834 . . 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 11913 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  RR )
2726recnd 8856 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( A  -  C ) )  e.  CC )
2827sqcld 11238 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( A  -  C )
) ^ 2 )  e.  CC )
296abscld 11913 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  RR )
3029recnd 8856 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( B  -  C ) )  e.  CC )
3130sqcld 11238 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( abs `  ( B  -  C )
) ^ 2 )  e.  CC )
3228, 31addcomd 9009 . . . 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 5829 . . . . 5  |-  ( X ^ 2 )  =  ( ( abs `  ( B  -  C )
) ^ 2 )
35 lawcos.3 . . . . . 6  |-  Y  =  ( abs `  ( A  -  C )
)
3635oveq1i 5829 . . . . 5  |-  ( Y ^ 2 )  =  ( ( abs `  ( A  -  C )
) ^ 2 )
3734, 36oveq12i 5831 . . . 4  |-  ( ( X ^ 2 )  +  ( Y ^
2 ) )  =  ( ( ( abs `  ( B  -  C
) ) ^ 2 )  +  ( ( abs `  ( A  -  C ) ) ^ 2 ) )
3832, 37syl6reqr 2334 . . 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 8851 . . . . . 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 5831 . . . . . 6  |-  ( X  x.  Y )  =  ( ( abs `  ( B  -  C )
)  x.  ( abs `  ( A  -  C
) ) )
4139, 40syl6reqr 2334 . . . . 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 5488 . . . . . . . 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 20094 . . . . . . . . . 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 5489 . . . . . . . 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 2327 . . . . . . 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 9531 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  e.  CC )
503, 6, 12, 18divne0d 9547 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
( A  -  C
)  /  ( B  -  C ) )  =/=  0 )
51 logcl 19921 . . . . . . . . . 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 11675 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  (
Im `  ( log `  ( ( A  -  C )  /  ( B  -  C )
) ) )  e.  RR )
54 recosval 12411 . . . . . . . 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 2315 . . . . . 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 19956 . . . . . . . 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 5489 . . . . . 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 11913 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  e.  RR )
6149, 50absne0d 11924 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( A  =/=  C  /\  B  =/=  C
) )  ->  ( abs `  ( ( A  -  C )  / 
( B  -  C
) ) )  =/=  0 )
6260, 49, 61redivd 11709 . . . . . 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 2319 . . . . 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 5837 . . . 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 5835 . . 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 5837 . 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 2325 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 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   ` cfv 5220  (class class class)co 5819    e. cmpt2 5821   CCcc 8730   RRcr 8731   0cc0 8732   _ici 8734    + caddc 8735    x. cmul 8737    - cmin 9032    / cdiv 9418   2c2 9790   ^cexp 11099   Recre 11577   Imcim 11578   abscabs 11714   expce 12338   cosccos 12341   logclog 19907
This theorem is referenced by:  pythag  20110  ssscongptld  20117
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 4186  ax-pr 4212  ax-un 4510  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
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 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-se 4351  df-we 4352  df-ord 4393  df-on 4394  df-lim 4395  df-suc 4396  df-om 4655  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-isom 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10655  df-ioc 10656  df-ico 10657  df-icc 10658  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-sum 12154  df-ef 12344  df-sin 12346  df-cos 12347  df-pi 12349  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-starv 13218  df-sca 13219  df-vsca 13220  df-tset 13222  df-ple 13223  df-ds 13225  df-hom 13227  df-cco 13228  df-rest 13322  df-topn 13323  df-topgen 13339  df-pt 13340  df-prds 13343  df-xrs 13398  df-0g 13399  df-gsum 13400  df-qtop 13405  df-imas 13406  df-xps 13408  df-mre 13483  df-mrc 13484  df-acs 13486  df-mnd 14362  df-submnd 14411  df-mulg 14487  df-cntz 14788  df-cmn 15086  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-cnfld 16373  df-top 16631  df-bases 16633  df-topon 16634  df-topsp 16635  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863  df-perf 16864  df-cn 16952  df-cnp 16953  df-haus 17038  df-tx 17252  df-hmeo 17441  df-fbas 17515  df-fg 17516  df-fil 17536  df-fm 17628  df-flim 17629  df-flf 17630  df-xms 17880  df-ms 17881  df-tms 17882  df-cncf 18377  df-limc 19211  df-dv 19212  df-log 19909
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