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Theorem angpieqvd 20055
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 20053 . . . . . 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 20054 . . . . . . 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 20052 . . . . 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 9090 . . . . . . . 8  |-  ( ph  ->  ( C  -  B
)  e.  CC )
1817adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  e.  CC )
194, 2subcld 9090 . . . . . . . 8  |-  ( ph  ->  ( C  -  A
)  e.  CC )
2019adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  e.  CC )
2114necomd 2502 . . . . . . . 8  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  C  =/=  A )
2211, 9, 21subne0d 9099 . . . . . . 7  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  A )  =/=  0 )
2318, 20, 22divcan1d 9470 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( ( C  -  B )  /  ( C  -  A )
)  x.  ( C  -  A ) )  =  ( C  -  B ) )
2423eqcomd 2261 . . . . 5  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  ( C  -  B )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  ( C  -  A
) ) )
2518, 20, 22divcld 9469 . . . . . 6  |-  ( (
ph  /\  ( ( A  -  B ) F ( C  -  B ) )  =  pi )  ->  (
( C  -  B
)  /  ( C  -  A ) )  e.  CC )
269, 10, 11, 25affineequiv 20050 . . . . 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 5764 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
w  x.  A )  =  ( ( ( C  -  B )  /  ( C  -  A ) )  x.  A ) )
29 oveq2 5765 . . . . . . . 8  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
1  -  w )  =  ( 1  -  ( ( C  -  B )  /  ( C  -  A )
) ) )
3029oveq1d 5772 . . . . . . 7  |-  ( w  =  ( ( C  -  B )  / 
( C  -  A
) )  ->  (
( 1  -  w
)  x.  C )  =  ( ( 1  -  ( ( C  -  B )  / 
( C  -  A
) ) )  x.  C ) )
3128, 30oveq12d 5775 . . . . . 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 2267 . . . . 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 2835 . . . 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 10617 . . . . . 6  |-  ( w  e.  ( 0 (,) 1 )  ->  w  e.  RR )
41 recn 8760 . . . . . 6  |-  ( w  e.  RR  ->  w  e.  CC )
4239, 40, 413syl 20 . . . . 5  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
) )  ->  w  e.  CC )
4336, 37, 38, 42affineequiv 20050 . . . 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 2502 . . . . . . . . . . . . . 14  |-  ( ph  ->  C  =/=  B )
494, 3, 48subne0d 9099 . . . . . . . . . . . . 13  |-  ( ph  ->  ( C  -  B
)  =/=  0 )
50493ad2ant1 981 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  B )  =/=  0
)
5144, 50eqnetrrd 2439 . . . . . . . . . . 11  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( w  x.  ( C  -  A
) )  =/=  0
)
5246, 47, 51mulne0bbd 9355 . . . . . . . . . 10  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  ( C  -  A )  =/=  0
)
5345, 46, 47, 52divmul3d 9503 . . . . . . . . 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 2330 . . . . . . 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 9101 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  C  =/=  A )
6261necomd 2502 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( 0 (,) 1
)  /\  ( C  -  B )  =  ( w  x.  ( C  -  A ) ) )  ->  A  =/=  C )
6357, 58, 59, 60, 62angpieqvdlem 20052 . . . . . . 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 20053 . . . . . 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 2638 . 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 2419   E.wrex 2517    \ cdif 3091   {csn 3581   ` cfv 4638  (class class class)co 5757    e. cmpt2 5759   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970   -ucneg 8971    / cdiv 9356   RR+crp 10286   (,)cioo 10587   Imcim 11513   picpi 12275   logclog 19839
This theorem is referenced by:  chordthm  20061
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841
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