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Theorem ssscongptld 23624
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 23618 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 23362 . . . . 5  |-  ( -u pi (,] pi )  C_  RR
2 ax-resscn 9595 . . . . 5  |-  RR  C_  CC
31, 2sstri 3479 . . . 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 9985 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
8 ssscongptld.7 . . . . . 6  |-  ( ph  ->  A  =/=  B )
95, 6, 8subne0d 9994 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
10 ssscongptld.3 . . . . . 6  |-  ( ph  ->  C  e.  CC )
1110, 6subcld 9985 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
12 ssscongptld.8 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
1312necomd 2702 . . . . . 6  |-  ( ph  ->  C  =/=  B )
1410, 6, 13subne0d 9994 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
154, 7, 9, 11, 14angcld 23607 . . . 4  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  ( -u pi (,] pi ) )
163, 15sseldi 3468 . . 3  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  CC )
1716coscld 14163 . 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 9985 . . . . 5  |-  ( ph  ->  ( D  -  E
)  e.  CC )
21 ssscongptld.9 . . . . . 6  |-  ( ph  ->  D  =/=  E )
2218, 19, 21subne0d 9994 . . . . 5  |-  ( ph  ->  ( D  -  E
)  =/=  0 )
23 ssscongptld.6 . . . . . 6  |-  ( ph  ->  G  e.  CC )
2423, 19subcld 9985 . . . . 5  |-  ( ph  ->  ( G  -  E
)  e.  CC )
25 ssscongptld.10 . . . . . . 7  |-  ( ph  ->  E  =/=  G )
2625necomd 2702 . . . . . 6  |-  ( ph  ->  G  =/=  E )
2723, 19, 26subne0d 9994 . . . . 5  |-  ( ph  ->  ( G  -  E
)  =/=  0 )
284, 20, 22, 24, 27angcld 23607 . . . 4  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  ( -u pi (,] pi ) )
293, 28sseldi 3468 . . 3  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  CC )
3029coscld 14163 . 2  |-  ( ph  ->  ( cos `  (
( D  -  E
) F ( G  -  E ) ) )  e.  CC )
3120abscld 13476 . . . 4  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  RR )
3231recnd 9668 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  CC )
3324abscld 13476 . . . 4  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  RR )
3433recnd 9668 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  CC )
3532, 34mulcld 9662 . 2  |-  ( ph  ->  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  e.  CC )
3620, 22absne0d 13487 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  =/=  0 )
3724, 27absne0d 13487 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =/=  0 )
3832, 34, 36, 37mulne0d 10263 . 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 13493 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( B  -  C
) ) )
4223, 19abssubd 13493 . . . . . 6  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =  ( abs `  ( E  -  G
) ) )
4340, 41, 423eqtr4d 2480 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( G  -  E
) ) )
4439, 43oveq12d 6323 . . . 4  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  =  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) ) )
4544oveq1d 6320 . . 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 2517 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  CC )
4743, 34eqeltrd 2517 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  e.  CC )
4846, 47mulcld 9662 . . . . 5  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  e.  CC )
4948, 17mulcld 9662 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  e.  CC )
5035, 30mulcld 9662 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) )  e.  CC )
51 2cnd 10682 . . . 4  |-  ( ph  ->  2  e.  CC )
52 2ne0 10702 . . . . 5  |-  2  =/=  0
5352a1i 11 . . . 4  |-  ( ph  ->  2  =/=  0 )
5432sqcld 12411 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( D  -  E )
) ^ 2 )  e.  CC )
5534sqcld 12411 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( G  -  E )
) ^ 2 )  e.  CC )
5654, 55addcld 9661 . . . . 5  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  e.  CC )
5751, 49mulcld 9662 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )  e.  CC )
5851, 50mulcld 9662 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )  e.  CC )
5939oveq1d 6320 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( abs `  ( D  -  E
) ) ^ 2 ) )
6043oveq1d 6320 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( C  -  B )
) ^ 2 )  =  ( ( abs `  ( G  -  E
) ) ^ 2 ) )
6159, 60oveq12d 6323 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) ) ^ 2 )  +  ( ( abs `  ( C  -  B ) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  E ) ) ^ 2 )  +  ( ( abs `  ( G  -  E )
) ^ 2 ) ) )
6261oveq1d 6320 . . . . . 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 ) ) ) ) ) ) )
63 ssscongptld.13 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( C  -  A )
)  =  ( abs `  ( G  -  D
) ) )
6463oveq1d 6320 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( C  -  A )
) ^ 2 )  =  ( ( abs `  ( G  -  D
) ) ^ 2 ) )
65 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( A  -  B
) )  =  ( abs `  ( A  -  B ) )
66 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( C  -  B
) )  =  ( abs `  ( C  -  B ) )
67 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( C  -  A
) )  =  ( abs `  ( C  -  A ) )
68 eqid 2429 . . . . . . . . 9  |-  ( ( A  -  B ) F ( C  -  B ) )  =  ( ( A  -  B ) F ( C  -  B ) )
694, 65, 66, 67, 68lawcos 23618 . . . . . . . 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 ) ) ) ) ) ) )
7010, 5, 6, 13, 8, 69syl32anc 1272 . . . . . . 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 ) ) ) ) ) ) )
71 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( D  -  E
) )  =  ( abs `  ( D  -  E ) )
72 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( G  -  E
) )  =  ( abs `  ( G  -  E ) )
73 eqid 2429 . . . . . . . . 9  |-  ( abs `  ( G  -  D
) )  =  ( abs `  ( G  -  D ) )
74 eqid 2429 . . . . . . . . 9  |-  ( ( D  -  E ) F ( G  -  E ) )  =  ( ( D  -  E ) F ( G  -  E ) )
754, 71, 72, 73, 74lawcos 23618 . . . . . . . 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 ) ) ) ) ) ) )
7623, 18, 19, 26, 21, 75syl32anc 1272 . . . . . . 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 ) ) ) ) ) ) )
7764, 70, 763eqtr3d 2478 . . . . . 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 ) ) ) ) ) ) )
7862, 77eqtr3d 2472 . . . . 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 ) ) ) ) ) ) )
7956, 57, 58, 78subcand 10026 . . . 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 ) ) ) ) ) )
8049, 50, 51, 53, 79mulcanad 10246 . . 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 ) ) ) ) )
8145, 80eqtr3d 2472 . 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 ) ) ) ) )
8217, 30, 35, 38, 81mulcanad 10246 1  |-  ( ph  ->  ( cos `  (
( A  -  B
) F ( C  -  B ) ) )  =  ( cos `  ( ( D  -  E ) F ( G  -  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439   {csn 4002   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   CCcc 9536   RRcr 9537   0cc0 9538    + caddc 9541    x. cmul 9543    - cmin 9859   -ucneg 9860    / cdiv 10268   2c2 10659   (,]cioc 11636   ^cexp 12269   Imcim 13140   abscabs 13276   cosccos 14095   picpi 14097   logclog 23377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-mulr 15167  df-starv 15168  df-sca 15169  df-vsca 15170  df-ip 15171  df-tset 15172  df-ple 15173  df-ds 15175  df-unif 15176  df-hom 15177  df-cco 15178  df-rest 15284  df-topn 15285  df-0g 15303  df-gsum 15304  df-topgen 15305  df-pt 15306  df-prds 15309  df-xrs 15363  df-qtop 15368  df-imas 15369  df-xps 15371  df-mre 15447  df-mrc 15448  df-acs 15450  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-submnd 16538  df-mulg 16631  df-cntz 16926  df-cmn 17371  df-psmet 18901  df-xmet 18902  df-met 18903  df-bl 18904  df-mopn 18905  df-fbas 18906  df-fg 18907  df-cnfld 18910  df-top 19856  df-bases 19857  df-topon 19858  df-topsp 19859  df-cld 19969  df-ntr 19970  df-cls 19971  df-nei 20049  df-lp 20087  df-perf 20088  df-cn 20178  df-cnp 20179  df-haus 20266  df-tx 20512  df-hmeo 20705  df-fil 20796  df-fm 20888  df-flim 20889  df-flf 20890  df-xms 21270  df-ms 21271  df-tms 21272  df-cncf 21810  df-limc 22706  df-dv 22707  df-log 23379
This theorem is referenced by:  chordthmlem  23631
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