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Theorem ssscongptld 22219
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 22211 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 21995 . . . . 5  |-  ( -u pi (,] pi )  C_  RR
2 ax-resscn 9338 . . . . 5  |-  RR  C_  CC
31, 2sstri 3364 . . . 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 9718 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  CC )
8 ssscongptld.7 . . . . . 6  |-  ( ph  ->  A  =/=  B )
95, 6, 8subne0d 9727 . . . . 5  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
10 ssscongptld.3 . . . . . 6  |-  ( ph  ->  C  e.  CC )
1110, 6subcld 9718 . . . . 5  |-  ( ph  ->  ( C  -  B
)  e.  CC )
12 ssscongptld.8 . . . . . . 7  |-  ( ph  ->  B  =/=  C )
1312necomd 2694 . . . . . 6  |-  ( ph  ->  C  =/=  B )
1410, 6, 13subne0d 9727 . . . . 5  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
154, 7, 9, 11, 14angcld 22200 . . . 4  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  ( -u pi (,] pi ) )
163, 15sseldi 3353 . . 3  |-  ( ph  ->  ( ( A  -  B ) F ( C  -  B ) )  e.  CC )
1716coscld 13414 . 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 9718 . . . . 5  |-  ( ph  ->  ( D  -  E
)  e.  CC )
21 ssscongptld.9 . . . . . 6  |-  ( ph  ->  D  =/=  E )
2218, 19, 21subne0d 9727 . . . . 5  |-  ( ph  ->  ( D  -  E
)  =/=  0 )
23 ssscongptld.6 . . . . . 6  |-  ( ph  ->  G  e.  CC )
2423, 19subcld 9718 . . . . 5  |-  ( ph  ->  ( G  -  E
)  e.  CC )
25 ssscongptld.10 . . . . . . 7  |-  ( ph  ->  E  =/=  G )
2625necomd 2694 . . . . . 6  |-  ( ph  ->  G  =/=  E )
2723, 19, 26subne0d 9727 . . . . 5  |-  ( ph  ->  ( G  -  E
)  =/=  0 )
284, 20, 22, 24, 27angcld 22200 . . . 4  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  ( -u pi (,] pi ) )
293, 28sseldi 3353 . . 3  |-  ( ph  ->  ( ( D  -  E ) F ( G  -  E ) )  e.  CC )
3029coscld 13414 . 2  |-  ( ph  ->  ( cos `  (
( D  -  E
) F ( G  -  E ) ) )  e.  CC )
3120abscld 12921 . . . 4  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  RR )
3231recnd 9411 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  e.  CC )
3324abscld 12921 . . . 4  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  RR )
3433recnd 9411 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  e.  CC )
3532, 34mulcld 9405 . 2  |-  ( ph  ->  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  e.  CC )
3620, 22absne0d 12932 . . 3  |-  ( ph  ->  ( abs `  ( D  -  E )
)  =/=  0 )
3724, 27absne0d 12932 . . 3  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =/=  0 )
3832, 34, 36, 37mulne0d 9987 . 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 12938 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( B  -  C
) ) )
4223, 19abssubd 12938 . . . . . 6  |-  ( ph  ->  ( abs `  ( G  -  E )
)  =  ( abs `  ( E  -  G
) ) )
4340, 41, 423eqtr4d 2484 . . . . 5  |-  ( ph  ->  ( abs `  ( C  -  B )
)  =  ( abs `  ( G  -  E
) ) )
4439, 43oveq12d 6108 . . . 4  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  =  ( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) ) )
4544oveq1d 6105 . . 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 2516 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  CC )
4743, 34eqeltrd 2516 . . . . . 6  |-  ( ph  ->  ( abs `  ( C  -  B )
)  e.  CC )
4846, 47mulcld 9405 . . . . 5  |-  ( ph  ->  ( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  e.  CC )
4948, 17mulcld 9405 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) )  x.  ( abs `  ( C  -  B ) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) )  e.  CC )
5035, 30mulcld 9405 . . . 4  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) )  x.  ( abs `  ( G  -  E ) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) )  e.  CC )
51 2cnd 10393 . . . 4  |-  ( ph  ->  2  e.  CC )
52 2ne0 10413 . . . . 5  |-  2  =/=  0
5352a1i 11 . . . 4  |-  ( ph  ->  2  =/=  0 )
5432sqcld 12005 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( D  -  E )
) ^ 2 )  e.  CC )
5534sqcld 12005 . . . . . 6  |-  ( ph  ->  ( ( abs `  ( G  -  E )
) ^ 2 )  e.  CC )
5654, 55addcld 9404 . . . . 5  |-  ( ph  ->  ( ( ( abs `  ( D  -  E
) ) ^ 2 )  +  ( ( abs `  ( G  -  E ) ) ^ 2 ) )  e.  CC )
5751, 49mulcld 9405 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( A  -  B )
)  x.  ( abs `  ( C  -  B
) ) )  x.  ( cos `  (
( A  -  B
) F ( C  -  B ) ) ) ) )  e.  CC )
5851, 50mulcld 9405 . . . . 5  |-  ( ph  ->  ( 2  x.  (
( ( abs `  ( D  -  E )
)  x.  ( abs `  ( G  -  E
) ) )  x.  ( cos `  (
( D  -  E
) F ( G  -  E ) ) ) ) )  e.  CC )
5939oveq1d 6105 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( abs `  ( D  -  E
) ) ^ 2 ) )
6043oveq1d 6105 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  ( C  -  B )
) ^ 2 )  =  ( ( abs `  ( G  -  E
) ) ^ 2 ) )
6159, 60oveq12d 6108 . . . . . . 7  |-  ( ph  ->  ( ( ( abs `  ( A  -  B
) ) ^ 2 )  +  ( ( abs `  ( C  -  B ) ) ^ 2 ) )  =  ( ( ( abs `  ( D  -  E ) ) ^ 2 )  +  ( ( abs `  ( G  -  E )
) ^ 2 ) ) )
6261oveq1d 6105 . . . . . 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 6105 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( C  -  A )
) ^ 2 )  =  ( ( abs `  ( G  -  D
) ) ^ 2 ) )
65 eqid 2442 . . . . . . . . 9  |-  ( abs `  ( A  -  B
) )  =  ( abs `  ( A  -  B ) )
66 eqid 2442 . . . . . . . . 9  |-  ( abs `  ( C  -  B
) )  =  ( abs `  ( C  -  B ) )
67 eqid 2442 . . . . . . . . 9  |-  ( abs `  ( C  -  A
) )  =  ( abs `  ( C  -  A ) )
68 eqid 2442 . . . . . . . . 9  |-  ( ( A  -  B ) F ( C  -  B ) )  =  ( ( A  -  B ) F ( C  -  B ) )
694, 65, 66, 67, 68lawcos 22211 . . . . . . . 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 1226 . . . . . . 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 2442 . . . . . . . . 9  |-  ( abs `  ( D  -  E
) )  =  ( abs `  ( D  -  E ) )
72 eqid 2442 . . . . . . . . 9  |-  ( abs `  ( G  -  E
) )  =  ( abs `  ( G  -  E ) )
73 eqid 2442 . . . . . . . . 9  |-  ( abs `  ( G  -  D
) )  =  ( abs `  ( G  -  D ) )
74 eqid 2442 . . . . . . . . 9  |-  ( ( D  -  E ) F ( G  -  E ) )  =  ( ( D  -  E ) F ( G  -  E ) )
754, 71, 72, 73, 74lawcos 22211 . . . . . . . 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 1226 . . . . . . 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 2482 . . . . . 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 2476 . . . . 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 9759 . . . 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 9970 . . 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 2476 . 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 9970 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 1369    e. wcel 1756    =/= wne 2605    \ cdif 3324   {csn 3876   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   CCcc 9279   RRcr 9280   0cc0 9281    + caddc 9284    x. cmul 9286    - cmin 9594   -ucneg 9595    / cdiv 9992   2c2 10370   (,]cioc 11300   ^cexp 11864   Imcim 12586   abscabs 12722   cosccos 13349   picpi 13351   logclog 22005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007
This theorem is referenced by:  chordthmlem  22226
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