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Theorem angpieqvd 20660
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 20658 . . . . . 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 20659 . . . . . . 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 20657 . . . . 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 9400 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1817adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  e.  CC )
194, 2subcld 9400 . . . . . . . 8  |-  ( ph  ->  ( C  -  A
)  e.  CC )
2019adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  e.  CC )
2114necomd 2681 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  =/=  A )
2211, 9, 21subne0d 9409 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  =/=  0 )
2318, 20, 22divcan1d 9780 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) )  =  ( C  -  B ) )
2423eqcomd 2440 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  ( C  -  A
) ) )
2518, 20, 22divcld 9779 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  CC )
269, 10, 11, 25affineequiv 20655 . . . . 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 6079 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
w  x.  A )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A ) )
29 oveq2 6080 . . . . . . . 8  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
1  -  w )  =  ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) ) )
3029oveq1d 6087 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( 1  -  w
)  x.  C )  =  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )
3128, 30oveq12d 6090 . . . . . 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 2446 . . . . 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 3044 . . . 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 10935 . . . . . 6  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  RR )
41 recn 9069 . . . . . 6  |-  ( w  e.  RR  ->  w  e.  CC )
4239, 40, 413syl 19 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  CC )
4336, 37, 38, 42affineequiv 20655 . . . 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 2681 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  B )
494, 3, 48subne0d 9409 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
50493ad2ant1 978 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =/=  0
)
5144, 50eqnetrrd 2618 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( w  x.  ( C  -  A
) )  =/=  0
)
5246, 47, 51mulne0bbd 9665 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  =/=  0
)
5345, 46, 47, 52divmul3d 9813 . . . . . . . . 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 2509 . . . . . . 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 9411 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  =/=  A )
6261necomd 2681 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  C )
6357, 58, 59, 60, 62angpieqvdlem 20657 . . . . . . 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 20658 . . . . . 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 2822 . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309   {csn 3806   ` cfv 5445  (class class class)co 6072    e. cmpt2 6074   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    - cmin 9280   -ucneg 9281    / cdiv 9666   RR+crp 10601   (,)cioo 10905   Imcim 11891   picpi 12657   logclog 20440
This theorem is referenced by:  chordthm  20666  chordthmALT  28900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442
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