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Theorem angpieqvd 20091
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 20089 . . . . . 6  |-  ( ph  ->  ( -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ 
<->  ( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
87biimpar 473 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  -u (
( C  -  B
)  /  ( A  -  B ) )  e.  RR+ )
92adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  e.  CC )
103adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  B  e.  CC )
114adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  e.  CC )
125adantr 453 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  =/=  B )
131, 2, 3, 4, 5, 6angpined 20090 . . . . . . 7  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi 
->  A  =/=  C
) )
1413imp 420 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  A  =/=  C )
159, 10, 11, 12, 14angpieqvdlem 20088 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( -u ( ( C  -  B )  /  ( A  -  B )
)  e.  RR+  <->  ( ( C  -  B )  /  ( C  -  A ) )  e.  ( 0 (,) 1
) ) )
168, 15mpbid 203 . . . 4  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  ( 0 (,) 1 ) )
174, 3subcld 9125 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1817adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  e.  CC )
194, 2subcld 9125 . . . . . . . 8  |-  ( ph  ->  ( C  -  A
)  e.  CC )
2019adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  e.  CC )
2114necomd 2504 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  =/=  A )
2211, 9, 21subne0d 9134 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  =/=  0 )
2318, 20, 22divcan1d 9505 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) )  =  ( C  -  B ) )
2423eqcomd 2263 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  ( C  -  A
) ) )
2518, 20, 22divcld 9504 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  CC )
269, 10, 11, 25affineequiv 20086 . . . . 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 225 . . . 4  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  B  =  ( ( ( ( C  -  B
)  /  ( C  -  A ) )  x.  A )  +  ( ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) )  x.  C
) ) )
28 oveq1 5799 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
w  x.  A )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A ) )
29 oveq2 5800 . . . . . . . 8  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
1  -  w )  =  ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) ) )
3029oveq1d 5807 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( 1  -  w
)  x.  C )  =  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )
3128, 30oveq12d 5810 . . . . . 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 2269 . . . . 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 ) ) ) )
3332rcla4ev 2859 . . . 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 645 . . 3  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  E. w  e.  ( 0 (,) 1
) B  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  C ) ) )
3534ex 425 . 2  |-  ( ph  ->  ( ( ( A  -  B ) F ( C  -  B
) )  =  pi 
->  E. w  e.  ( 0 (,) 1 ) B  =  ( ( w  x.  A )  +  ( ( 1  -  w )  x.  C ) ) ) )
362adantr 453 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  A  e.  CC )
373adantr 453 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  B  e.  CC )
384adantr 453 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  C  e.  CC )
39 simpr 449 . . . . . 6  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  ( 0 (,) 1
) )
40 elioore 10653 . . . . . 6  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  RR )
41 recn 8795 . . . . . 6  |-  ( w  e.  RR  ->  w  e.  CC )
4239, 40, 413syl 20 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  CC )
4336, 37, 38, 42affineequiv 20086 . . . 4  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  ( B  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  C ) )  <->  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) ) )
44 simp3 962 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )
45173ad2ant1 981 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  e.  CC )
46423adant3 980 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  w  e.  CC )
47193ad2ant1 981 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  e.  CC )
486necomd 2504 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  B )
494, 3, 48subne0d 9134 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
50493ad2ant1 981 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =/=  0
)
5144, 50eqnetrrd 2441 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( w  x.  ( C  -  A
) )  =/=  0
)
5246, 47, 51mulne0bbd 9390 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  =/=  0
)
5345, 46, 47, 52divmul3d 9538 . . . . . . . . 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 225 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( C  -  B )  /  ( C  -  A ) )  =  w )
55 simp2 961 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  w  e.  ( 0 (,) 1
) )
5654, 55eqeltrd 2332 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( C  -  B )  /  ( C  -  A ) )  e.  ( 0 (,) 1
) )
5723ad2ant1 981 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  e.  CC )
5833ad2ant1 981 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  B  e.  CC )
5943ad2ant1 981 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  e.  CC )
6053ad2ant1 981 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  B )
6159, 57, 52subne0ad 9136 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  =/=  A )
6261necomd 2504 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  C )
6357, 58, 59, 60, 62angpieqvdlem 20088 . . . . . . 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 225 . . . . . 6  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  -u ( ( C  -  B )  /  ( A  -  B ) )  e.  RR+ )
6563ad2ant1 981 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  B  =/=  C )
661, 57, 58, 59, 60, 65angpieqvdlem2 20089 . . . . . 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 203 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( ( A  -  B ) F ( C  -  B ) )  =  pi )
68673expia 1158 . . . 4  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  (
( C  -  B
)  =  ( w  x.  ( C  -  A ) )  -> 
( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
6943, 68sylbid 208 . . 3  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  ( B  =  ( (
w  x.  A )  +  ( ( 1  -  w )  x.  C ) )  -> 
( ( A  -  B ) F ( C  -  B ) )  =  pi ) )
7069rexlimdva 2642 . 2  |-  ( ph  ->  ( E. w  e.  ( 0 (,) 1
) B  =  ( ( w  x.  A
)  +  ( ( 1  -  w )  x.  C ) )  ->  ( ( A  -  B ) F ( C  -  B
) )  =  pi ) )
7135, 70impbid 185 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519    \ cdif 3124   {csn 3614   ` cfv 4673  (class class class)co 5792    e. cmpt2 5794   CCcc 8703   RRcr 8704   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006    / cdiv 9391   RR+crp 10322   (,)cioo 10623   Imcim 11549   picpi 12311   logclog 19875
This theorem is referenced by:  chordthm  20097
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ioc 10628  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-shft 11528  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-limsup 11911  df-clim 11928  df-rlim 11929  df-sum 12125  df-ef 12312  df-sin 12314  df-cos 12315  df-pi 12317  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-lp 16831  df-perf 16832  df-cn 16920  df-cnp 16921  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cncf 18345  df-limc 19179  df-dv 19180  df-log 19877
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