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Theorem angpieqvd 19872
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 19870 . . . . . 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 19871 . . . . . . 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 19869 . . . . 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 9037 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1817adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  e.  CC )
194, 2subcld 9037 . . . . . . . 8  |-  ( ph  ->  ( C  -  A
)  e.  CC )
2019adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  e.  CC )
2114necomd 2495 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  =/=  A )
2211, 9, 21subne0d 9046 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  =/=  0 )
2318, 20, 22divcan1d 9417 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) )  =  ( C  -  B ) )
2423eqcomd 2258 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  ( C  -  A
) ) )
2518, 20, 22divcld 9416 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  CC )
269, 10, 11, 25affineequiv 19867 . . . . 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 5717 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
w  x.  A )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A ) )
29 oveq2 5718 . . . . . . . 8  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
1  -  w )  =  ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) ) )
3029oveq1d 5725 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( 1  -  w
)  x.  C )  =  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )
3128, 30oveq12d 5728 . . . . . 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 2264 . . . . 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 2821 . . . 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 10564 . . . . . 6  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  RR )
41 recn 8707 . . . . . 6  |-  ( w  e.  RR  ->  w  e.  CC )
4239, 40, 413syl 20 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  CC )
4336, 37, 38, 42affineequiv 19867 . . . 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 2495 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  B )
494, 3, 48subne0d 9046 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
50493ad2ant1 981 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =/=  0
)
5144, 50eqnetrrd 2432 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( w  x.  ( C  -  A
) )  =/=  0
)
5246, 47, 51mulne0bbd 9302 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  =/=  0
)
5345, 46, 47, 52divmul3d 9450 . . . . . . . . 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 2327 . . . . . . 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 9048 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  =/=  A )
6261necomd 2495 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  C )
6357, 58, 59, 60, 62angpieqvdlem 19869 . . . . . . 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 19870 . . . . . 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 2629 . 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 2412   E.wrex 2510    \ cdif 3075   {csn 3544   ` cfv 4592  (class class class)co 5710    e. cmpt2 5712   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   RR+crp 10233   (,)cioo 10534   Imcim 11460   picpi 12222   logclog 19744
This theorem is referenced by:  chordthm  19878
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-fi 7049  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-ioo 10538  df-ioc 10539  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223  df-sin 12225  df-cos 12226  df-pi 12228  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-rest 13201  df-topn 13202  df-topgen 13218  df-pt 13219  df-prds 13222  df-xrs 13277  df-0g 13278  df-gsum 13279  df-qtop 13284  df-imas 13285  df-xps 13287  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-mulg 14327  df-cntz 14628  df-cmn 14926  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cld 16588  df-ntr 16589  df-cls 16590  df-nei 16667  df-lp 16700  df-perf 16701  df-cn 16789  df-cnp 16790  df-haus 16875  df-tx 17089  df-hmeo 17278  df-fbas 17352  df-fg 17353  df-fil 17373  df-fm 17465  df-flim 17466  df-flf 17467  df-xms 17717  df-ms 17718  df-tms 17719  df-cncf 18214  df-limc 19048  df-dv 19049  df-log 19746
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