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Theorem angpieqvd 20540
Description: The angle ABC is  pi iff B is a nontrivial convex combination of A and C, i.e., iff B is in the interior of the segment AC. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
angpieqvd.angdef  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
angpieqvd.A  |-  ( ph  ->  A  e.  CC )
angpieqvd.B  |-  ( ph  ->  B  e.  CC )
angpieqvd.C  |-  ( ph  ->  C  e.  CC )
angpieqvd.AneB  |-  ( ph  ->  A  =/=  B )
angpieqvd.BneC  |-  ( ph  ->  B  =/=  C )
Assertion
Ref Expression
angpieqvd  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C
) ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    w, F    ph, w    w, A    w, B    w, C
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem angpieqvd
StepHypRef Expression
1 angpieqvd.angdef . . . . . . 7  |-  F  =  ( x  e.  ( CC  \  { 0 } ) ,  y  e.  ( CC  \  { 0 } ) 
|->  ( Im `  ( log `  ( y  /  x ) ) ) )
2 angpieqvd.A . . . . . . 7  |-  ( ph  ->  A  e.  CC )
3 angpieqvd.B . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4 angpieqvd.C . . . . . . 7  |-  ( ph  ->  C  e.  CC )
5 angpieqvd.AneB . . . . . . 7  |-  ( ph  ->  A  =/=  B )
6 angpieqvd.BneC . . . . . . 7  |-  ( ph  ->  B  =/=  C )
71, 2, 3, 4, 5, 6angpieqvdlem2 20538 . . . . . 6  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
87biimpar 472 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  -u (
( C  -  B
)  /  ( A  -  B ) )  e.  RR+ )
92adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  e.  CC )
103adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  B  e.  CC )
114adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  e.  CC )
125adantr 452 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  =/=  B )
131, 2, 3, 4, 5, 6angpined 20539 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi 
->  A  =/=  C
) )
1413imp 419 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  =/=  C )
159, 10, 11, 12, 14angpieqvdlem 20537 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( -u ( ( C  -  B )  /  ( A  -  B )
)  e.  RR+  <->  ( ( C  -  B )  /  ( C  -  A ) )  e.  ( 0 (,) 1
) ) )
168, 15mpbid 202 . . . 4  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  ( 0 (,) 1 ) )
174, 3subcld 9344 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1817adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  e.  CC )
194, 2subcld 9344 . . . . . . . 8  |-  ( ph  ->  ( C  -  A
)  e.  CC )
2019adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  e.  CC )
2114necomd 2634 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  =/=  A )
2211, 9, 21subne0d 9353 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  =/=  0 )
2318, 20, 22divcan1d 9724 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) )  =  ( C  -  B ) )
2423eqcomd 2393 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  ( C  -  A
) ) )
2518, 20, 22divcld 9723 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  CC )
269, 10, 11, 25affineequiv 20535 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( B  =  ( (
( ( C  -  B )  /  ( C  -  A )
)  x.  A )  +  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )  <->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) ) ) )
2724, 26mpbird 224 . . . 4  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  B  =  ( ( ( ( C  -  B
)  /  ( C  -  A ) )  x.  A )  +  ( ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) )  x.  C
) ) )
28 oveq1 6028 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
w  x.  A )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A ) )
29 oveq2 6029 . . . . . . . 8  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
1  -  w )  =  ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) ) )
3029oveq1d 6036 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( 1  -  w
)  x.  C )  =  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )
3128, 30oveq12d 6039 . . . . . 6  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( w  x.  A
)  +  ( ( 1  -  w )  x.  C ) )  =  ( ( ( ( C  -  B
)  /  ( C  -  A ) )  x.  A )  +  ( ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) )  x.  C
) ) )
3231eqeq2d 2399 . . . . 5  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  ( B  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  C ) )  <->  B  =  ( ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A )  +  ( ( 1  -  (
( C  -  B
)  /  ( C  -  A ) ) )  x.  C ) ) ) )
3332rspcev 2996 . . . 4  |-  ( ( ( ( C  -  B )  /  ( C  -  A )
)  e.  ( 0 (,) 1 )  /\  B  =  ( (
( ( C  -  B )  /  ( C  -  A )
)  x.  A )  +  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) ) )  ->  E. w  e.  ( 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C ) ) )
3416, 27, 33syl2anc 643 . . 3  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  E. w  e.  ( 0 (,) 1
) B  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  C ) ) )
3534ex 424 . 2  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi 
->  E. w  e.  ( 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C ) ) ) )
362adantr 452 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  A  e.  CC )
373adantr 452 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  B  e.  CC )
384adantr 452 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  C  e.  CC )
39 simpr 448 . . . . . 6  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  ( 0 (,) 1
) )
40 elioore 10879 . . . . . 6  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  RR )
41 recn 9014 . . . . . 6  |-  ( w  e.  RR  ->  w  e.  CC )
4239, 40, 413syl 19 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  CC )
4336, 37, 38, 42affineequiv 20535 . . . 4  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  ( B  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  C ) )  <->  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) ) )
44 simp3 959 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )
45173ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  e.  CC )
46423adant3 977 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  w  e.  CC )
47193ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  e.  CC )
486necomd 2634 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  B )
494, 3, 48subne0d 9353 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
50493ad2ant1 978 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =/=  0
)
5144, 50eqnetrrd 2571 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( w  x.  ( C  -  A
) )  =/=  0
)
5246, 47, 51mulne0bbd 9609 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  =/=  0
)
5345, 46, 47, 52divmul3d 9757 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( (
( C  -  B
)  /  ( C  -  A ) )  =  w  <->  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) ) )
5444, 53mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( C  -  B )  /  ( C  -  A ) )  =  w )
55 simp2 958 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  w  e.  ( 0 (,) 1
) )
5654, 55eqeltrd 2462 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( C  -  B )  /  ( C  -  A ) )  e.  ( 0 (,) 1
) )
5723ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  e.  CC )
5833ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  B  e.  CC )
5943ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  e.  CC )
6053ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  B )
6159, 57, 52subne0ad 9355 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  =/=  A )
6261necomd 2634 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  C )
6357, 58, 59, 60, 62angpieqvdlem 20537 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( -u (
( C  -  B
)  /  ( A  -  B ) )  e.  RR+  <->  ( ( C  -  B )  / 
( C  -  A
) )  e.  ( 0 (,) 1 ) ) )
6456, 63mpbird 224 . . . . . 6  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ )
6563ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  B  =/=  C )
661, 57, 58, 59, 60, 65angpieqvdlem2 20538 . . . . . 6  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( -u (
( C  -  B
)  /  ( A  -  B ) )  e.  RR+  <->  ( ( A  -  B ) F ( C  -  B
) )  =  pi ) )
6764, 66mpbid 202 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( A  -  B ) F ( C  -  B ) )  =  pi )
68673expia 1155 . . . 4  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  (
( C  -  B
)  =  ( w  x.  ( C  -  A ) )  -> 
( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
6943, 68sylbid 207 . . 3  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  ( B  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  C ) )  -> 
( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
7069rexlimdva 2774 . 2  |-  ( ph  ->  ( E. w  e.  ( 0 (,) 1
) B  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  C ) )  ->  ( ( A  -  B ) F ( C  -  B
) )  =  pi ) )
7135, 70impbid 184 1  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi  <->  E. w  e.  ( 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651    \ cdif 3261   {csn 3758   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    - cmin 9224   -ucneg 9225    / cdiv 9610   RR+crp 10545   (,)cioo 10849   Imcim 11831   picpi 12597   logclog 20320
This theorem is referenced by:  chordthm  20546  chordthmALT  28388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322
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