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Theorem ssscongptld 20070
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 20062 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 19850 . . . . 5  |-  ( -u pi (,] pi )  C_  RR
2 ax-resscn 8748 . . . . 5  |-  RR  C_  CC
31, 2sstri 3149 . . . 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 9111 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
8 ssscongptld.7 . . . . . 6  |-  ( ph  ->  A  =/=  B )
95, 6, 8subne0d 9120 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
10 ssscongptld.3 . . . . . 6  |-  ( ph  ->  C  e.  CC )
1110, 6subcld 9111 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
12 ssscongptld.8 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
1312necomd 2502 . . . . . 6  |-  ( ph  ->  C  =/=  B )
1410, 6, 13subne0d 9120 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
154, 7, 9, 11, 14angcld 20051 . . . 4  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  ( -u pi (,] pi ) )
163, 15sseldi 3139 . . 3  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  CC )
1716coscld 12359 . 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 9111 . . . . 5  |-  ( ph  ->  ( D  -  E
)  e.  CC )
21 ssscongptld.9 . . . . . 6  |-  ( ph  ->  D  =/=  E )
2218, 19, 21subne0d 9120 . . . . 5  |-  ( ph  ->  ( D  -  E
)  =/=  0 )
23 ssscongptld.6 . . . . . 6  |-  ( ph  ->  G  e.  CC )
2423, 19subcld 9111 . . . . 5  |-  ( ph  ->  ( G  -  E
)  e.  CC )
25 ssscongptld.10 . . . . . . 7  |-  ( ph  ->  E  =/=  G )
2625necomd 2502 . . . . . 6  |-  ( ph  ->  G  =/=  E )
2723, 19, 26subne0d 9120 . . . . 5  |-  ( ph  ->  ( G  -  E
)  =/=  0 )
284, 20, 22, 24, 27angcld 20051 . . . 4  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  ( -u pi (,] pi ) )
293, 28sseldi 3139 . . 3  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  CC )
3029coscld 12359 . 2  |-  ( ph  ->  ( cos `  (
( D  -  E
) F ( G  -  E ) ) )  e.  CC )
3120abscld 11869 . . . 4  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  RR )
3231recnd 8815 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  CC )
3324abscld 11869 . . . 4  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  RR )
3433recnd 8815 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  CC )
3532, 34mulcld 8809 . 2  |-  ( ph  ->  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  e.  CC )
3620, 22absne0d 11880 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  =/=  0 )
3724, 27absne0d 11880 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =/=  0 )
3832, 34, 36, 37mulne0d 9374 . 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 11886 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( B  -  C
) ) )
4223, 19abssubd 11886 . . . . . 6  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =  ( abs `  ( E  -  G
) ) )
4340, 41, 423eqtr4d 2298 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( G  -  E
) ) )
4439, 43oveq12d 5796 . . . 4  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  =  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) ) )
4544oveq1d 5793 . . 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 8809 . . . . 5  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  e.  CC )
4948, 17mulcld 8809 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  e.  CC )
5035, 30mulcld 8809 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) )  e.  CC )
51 2cn 9770 . . . . 5  |-  2  e.  CC
5251a1i 12 . . . 4  |-  ( ph  ->  2  e.  CC )
53 2ne0 9783 . . . . 5  |-  2  =/=  0
5453a1i 12 . . . 4  |-  ( ph  ->  2  =/=  0 )
5532sqcld 11195 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( D  -  E )
) ^ 2 )  e.  CC )
5634sqcld 11195 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( G  -  E )
) ^ 2 )  e.  CC )
5755, 56addcld 8808 . . . . 5  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  e.  CC )
5852, 49mulcld 8809 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )  e.  CC )
5952, 50mulcld 8809 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )  e.  CC )
6039oveq1d 5793 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( abs `  ( D  -  E
) ) ^ 2 ) )
6143oveq1d 5793 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( C  -  B )
) ^ 2 )  =  ( ( abs `  ( G  -  E
) ) ^ 2 ) )
6260, 61oveq12d 5796 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) ) ^ 2 )  +  ( ( abs `  ( C  -  B ) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  E ) ) ^ 2 )  +  ( ( abs `  ( G  -  E )
) ^ 2 ) ) )
6362oveq1d 5793 . . . . . 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 5793 . . . . . . 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 20062 . . . . . . . 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 20062 . . . . . . . 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 9152 . . . 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 9357 . . 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 9357 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 3110   {csn 3600   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   CCcc 8689   RRcr 8690   0cc0 8691    + caddc 8694    x. cmul 8696    - cmin 8991   -ucneg 8992    / cdiv 9377   2c2 9749   (,]cioc 10609   ^cexp 11056   Imcim 11534   abscabs 11670   cosccos 12294   picpi 12296   logclog 19860
This theorem is referenced by:  chordthmlem  20077
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862
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