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Theorem ssscongptld 20049
Description: If two triangles have equal sides, one angle in one triangle has the same cosine as the corresponding angle in the other triangle. This is a partial form of the SSS congruence theorem.

This theorem is proven by using lawcos 20041 on both triangles to express one side in terms of the other two, and then equating these expressions and reducing this algebraically to get an equality of cosines of angles. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses
Ref Expression
ssscongptld.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
ssscongptld.1  |-  ( ph  ->  A  e.  CC )
ssscongptld.2  |-  ( ph  ->  B  e.  CC )
ssscongptld.3  |-  ( ph  ->  C  e.  CC )
ssscongptld.4  |-  ( ph  ->  D  e.  CC )
ssscongptld.5  |-  ( ph  ->  E  e.  CC )
ssscongptld.6  |-  ( ph  ->  G  e.  CC )
ssscongptld.7  |-  ( ph  ->  A  =/=  B )
ssscongptld.8  |-  ( ph  ->  B  =/=  C )
ssscongptld.9  |-  ( ph  ->  D  =/=  E )
ssscongptld.10  |-  ( ph  ->  E  =/=  G )
ssscongptld.11  |-  ( ph  ->  ( abs `  ( A  -  B )
)  =  ( abs `  ( D  -  E
) ) )
ssscongptld.12  |-  ( ph  ->  ( abs `  ( B  -  C )
)  =  ( abs `  ( E  -  G
) ) )
ssscongptld.13  |-  ( ph  ->  ( abs `  ( C  -  A )
)  =  ( abs `  ( G  -  D
) ) )
Assertion
Ref Expression
ssscongptld  |-  ( ph  ->  ( cos `  (
( A  -  B
) F ( C  -  B ) ) )  =  ( cos `  ( ( D  -  E ) F ( G  -  E ) ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y   
x, E, y    x, G, y
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem ssscongptld
StepHypRef Expression
1 negpitopissre 19829 . . . . 5  |-  ( -u pi (,] pi )  C_  RR
2 ax-resscn 8727 . . . . 5  |-  RR  C_  CC
31, 2sstri 3130 . . . 4  |-  ( -u pi (,] pi )  C_  CC
4 ssscongptld.angdef . . . . 5  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
5 ssscongptld.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
6 ssscongptld.2 . . . . . 6  |-  ( ph  ->  B  e.  CC )
75, 6subcld 9090 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
8 ssscongptld.7 . . . . . 6  |-  ( ph  ->  A  =/=  B )
95, 6, 8subne0d 9099 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
10 ssscongptld.3 . . . . . 6  |-  ( ph  ->  C  e.  CC )
1110, 6subcld 9090 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
12 ssscongptld.8 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
1312necomd 2502 . . . . . 6  |-  ( ph  ->  C  =/=  B )
1410, 6, 13subne0d 9099 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
154, 7, 9, 11, 14angcld 20030 . . . 4  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  ( -u pi (,] pi ) )
163, 15sseldi 3120 . . 3  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  CC )
1716coscld 12338 . 2  |-  ( ph  ->  ( cos `  (
( A  -  B
) F ( C  -  B ) ) )  e.  CC )
18 ssscongptld.4 . . . . . 6  |-  ( ph  ->  D  e.  CC )
19 ssscongptld.5 . . . . . 6  |-  ( ph  ->  E  e.  CC )
2018, 19subcld 9090 . . . . 5  |-  ( ph  ->  ( D  -  E
)  e.  CC )
21 ssscongptld.9 . . . . . 6  |-  ( ph  ->  D  =/=  E )
2218, 19, 21subne0d 9099 . . . . 5  |-  ( ph  ->  ( D  -  E
)  =/=  0 )
23 ssscongptld.6 . . . . . 6  |-  ( ph  ->  G  e.  CC )
2423, 19subcld 9090 . . . . 5  |-  ( ph  ->  ( G  -  E
)  e.  CC )
25 ssscongptld.10 . . . . . . 7  |-  ( ph  ->  E  =/=  G )
2625necomd 2502 . . . . . 6  |-  ( ph  ->  G  =/=  E )
2723, 19, 26subne0d 9099 . . . . 5  |-  ( ph  ->  ( G  -  E
)  =/=  0 )
284, 20, 22, 24, 27angcld 20030 . . . 4  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  ( -u pi (,] pi ) )
293, 28sseldi 3120 . . 3  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  CC )
3029coscld 12338 . 2  |-  ( ph  ->  ( cos `  (
( D  -  E
) F ( G  -  E ) ) )  e.  CC )
3120abscld 11848 . . . 4  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  RR )
3231recnd 8794 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  CC )
3324abscld 11848 . . . 4  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  RR )
3433recnd 8794 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  CC )
3532, 34mulcld 8788 . 2  |-  ( ph  ->  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  e.  CC )
3620, 22absne0d 11859 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  =/=  0 )
3724, 27absne0d 11859 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =/=  0 )
3832, 34, 36, 37mulne0d 9353 . 2  |-  ( ph  ->  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  =/=  0 )
39 ssscongptld.11 . . . . 5  |-  ( ph  ->  ( abs `  ( A  -  B )
)  =  ( abs `  ( D  -  E
) ) )
40 ssscongptld.12 . . . . . 6  |-  ( ph  ->  ( abs `  ( B  -  C )
)  =  ( abs `  ( E  -  G
) ) )
4110, 6abssubd 11865 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( B  -  C
) ) )
4223, 19abssubd 11865 . . . . . 6  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =  ( abs `  ( E  -  G
) ) )
4340, 41, 423eqtr4d 2298 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( G  -  E
) ) )
4439, 43oveq12d 5775 . . . 4  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  =  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) ) )
4544oveq1d 5772 . . 3  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  =  ( ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )
4639, 32eqeltrd 2330 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  CC )
4743, 34eqeltrd 2330 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  e.  CC )
4846, 47mulcld 8788 . . . . 5  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  e.  CC )
4948, 17mulcld 8788 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  e.  CC )
5035, 30mulcld 8788 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) )  e.  CC )
51 2cn 9749 . . . . 5  |-  2  e.  CC
5251a1i 12 . . . 4  |-  ( ph  ->  2  e.  CC )
53 2ne0 9762 . . . . 5  |-  2  =/=  0
5453a1i 12 . . . 4  |-  ( ph  ->  2  =/=  0 )
5532sqcld 11174 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( D  -  E )
) ^ 2 )  e.  CC )
5634sqcld 11174 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( G  -  E )
) ^ 2 )  e.  CC )
5755, 56addcld 8787 . . . . 5  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  e.  CC )
5852, 49mulcld 8788 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )  e.  CC )
5952, 50mulcld 8788 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )  e.  CC )
6039oveq1d 5772 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( abs `  ( D  -  E
) ) ^ 2 ) )
6143oveq1d 5772 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( C  -  B )
) ^ 2 )  =  ( ( abs `  ( G  -  E
) ) ^ 2 ) )
6260, 61oveq12d 5775 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) ) ^ 2 )  +  ( ( abs `  ( C  -  B ) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  E ) ) ^ 2 )  +  ( ( abs `  ( G  -  E )
) ^ 2 ) ) )
6362oveq1d 5772 . . . . . 6  |-  ( ph  ->  ( ( ( ( abs `  ( A  -  B ) ) ^ 2 )  +  ( ( abs `  ( C  -  B )
) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( A  -  B ) )  x.  ( abs `  ( C  -  B )
) )  x.  ( cos `  ( ( A  -  B ) F ( C  -  B
) ) ) ) ) )  =  ( ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) ) ) )
64 ssscongptld.13 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( C  -  A )
)  =  ( abs `  ( G  -  D
) ) )
6564oveq1d 5772 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( C  -  A )
) ^ 2 )  =  ( ( abs `  ( G  -  D
) ) ^ 2 ) )
66 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( A  -  B
) )  =  ( abs `  ( A  -  B ) )
67 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( C  -  B
) )  =  ( abs `  ( C  -  B ) )
68 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( C  -  A
) )  =  ( abs `  ( C  -  A ) )
69 eqid 2256 . . . . . . . . 9  |-  ( ( A  -  B ) F ( C  -  B ) )  =  ( ( A  -  B ) F ( C  -  B ) )
704, 66, 67, 68, 69lawcos 20041 . . . . . . . 8  |-  ( ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  /\  ( C  =/=  B  /\  A  =/=  B
) )  ->  (
( abs `  ( C  -  A )
) ^ 2 )  =  ( ( ( ( abs `  ( A  -  B )
) ^ 2 )  +  ( ( abs `  ( C  -  B
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) ) ) )
7110, 5, 6, 13, 8, 70syl32anc 1195 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( C  -  A )
) ^ 2 )  =  ( ( ( ( abs `  ( A  -  B )
) ^ 2 )  +  ( ( abs `  ( C  -  B
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) ) ) )
72 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( D  -  E
) )  =  ( abs `  ( D  -  E ) )
73 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( G  -  E
) )  =  ( abs `  ( G  -  E ) )
74 eqid 2256 . . . . . . . . 9  |-  ( abs `  ( G  -  D
) )  =  ( abs `  ( G  -  D ) )
75 eqid 2256 . . . . . . . . 9  |-  ( ( D  -  E ) F ( G  -  E ) )  =  ( ( D  -  E ) F ( G  -  E ) )
764, 72, 73, 74, 75lawcos 20041 . . . . . . . 8  |-  ( ( ( G  e.  CC  /\  D  e.  CC  /\  E  e.  CC )  /\  ( G  =/=  E  /\  D  =/=  E
) )  ->  (
( abs `  ( G  -  D )
) ^ 2 )  =  ( ( ( ( abs `  ( D  -  E )
) ^ 2 )  +  ( ( abs `  ( G  -  E
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) ) ) )
7723, 18, 19, 26, 21, 76syl32anc 1195 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( G  -  D )
) ^ 2 )  =  ( ( ( ( abs `  ( D  -  E )
) ^ 2 )  +  ( ( abs `  ( G  -  E
) ) ^ 2 ) )  -  (
2  x.  ( ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) ) ) )
7865, 71, 773eqtr3d 2296 . . . . . 6  |-  ( ph  ->  ( ( ( ( abs `  ( A  -  B ) ) ^ 2 )  +  ( ( abs `  ( C  -  B )
) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( A  -  B ) )  x.  ( abs `  ( C  -  B )
) )  x.  ( cos `  ( ( A  -  B ) F ( C  -  B
) ) ) ) ) )  =  ( ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) ) ) )
7963, 78eqtr3d 2290 . . . . 5  |-  ( ph  ->  ( ( ( ( abs `  ( D  -  E ) ) ^ 2 )  +  ( ( abs `  ( G  -  E )
) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( A  -  B ) )  x.  ( abs `  ( C  -  B )
) )  x.  ( cos `  ( ( A  -  B ) F ( C  -  B
) ) ) ) ) )  =  ( ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  -  ( 2  x.  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) ) ) )
8057, 58, 59, 79subcand 9131 . . . 4  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )  =  ( 2  x.  (
( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) ) )
8149, 50, 52, 54, 80mulcanad 9336 . . 3  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  =  ( ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )
8245, 81eqtr3d 2290 . 2  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  =  ( ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )
8317, 30, 35, 38, 82mulcanad 9336 1  |-  ( ph  ->  ( cos `  (
( A  -  B
) F ( C  -  B ) ) )  =  ( cos `  ( ( D  -  E ) F ( G  -  E ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2419    \ cdif 3091   {csn 3581   ` cfv 4638  (class class class)co 5757    e. cmpt2 5759   CCcc 8668   RRcr 8669   0cc0 8670    + caddc 8673    x. cmul 8675    - cmin 8970   -ucneg 8971    / cdiv 9356   2c2 9728   (,]cioc 10588   ^cexp 11035   Imcim 11513   abscabs 11649   cosccos 12273   picpi 12275   logclog 19839
This theorem is referenced by:  chordthmlem  20056
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841
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