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Theorem pntleme 21307
Description: Lemma for pnt 21313. Package up pntlemo 21306 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
Hypotheses
Ref Expression
pntlem1.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntlem1.a  |-  ( ph  ->  A  e.  RR+ )
pntlem1.b  |-  ( ph  ->  B  e.  RR+ )
pntlem1.l  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
pntlem1.d  |-  D  =  ( A  +  1 )
pntlem1.f  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
pntlem1.u  |-  ( ph  ->  U  e.  RR+ )
pntlem1.u2  |-  ( ph  ->  U  <_  A )
pntlem1.e  |-  E  =  ( U  /  D
)
pntlem1.k  |-  K  =  ( exp `  ( B  /  E ) )
pntlem1.y  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
pntlem1.x  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
pntlem1.c  |-  ( ph  ->  C  e.  RR+ )
pntlem1.w  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
pntleme.U  |-  ( ph  ->  A. z  e.  ( Y [,)  +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
pntleme.K  |-  ( ph  ->  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  <  z  /\  (
( 1  +  ( L  x.  E ) )  x.  z )  <  ( k  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
pntleme.C  |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo )
( ( ( ( abs `  ( R `
 z ) )  x.  ( log `  z
) )  -  (
( 2  /  ( log `  z ) )  x.  sum_ i  e.  ( 1 ... ( |_
`  ( z  /  Y ) ) ) ( ( abs `  ( R `  ( z  /  i ) ) )  x.  ( log `  i ) ) ) )  /  z )  <_  C )
Assertion
Ref Expression
pntleme  |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
Distinct variable groups:    z, C    w, F    y, z    u, k, y, z, L    k, K, y, z    ph, v    i, k, u, v, w, y, z, R    w, U, z    v, W, w, z    k, X, y, z    i, Y, z   
k, a, u, v, y, z, E
Allowed substitution hints:    ph( y, z, w, u, i, k, a)    A( y, z, w, v, u, i, k, a)    B( y, z, w, v, u, i, k, a)    C( y, w, v, u, i, k, a)    D( y, z, w, v, u, i, k, a)    R( a)    U( y, v, u, i, k, a)    E( w, i)    F( y, z, v, u, i, k, a)    K( w, v, u, i, a)    L( w, v, i, a)    W( y, u, i, k, a)    X( w, v, u, i, a)    Y( y, w, v, u, k, a)

Proof of Theorem pntleme
StepHypRef Expression
1 pntlem1.r . . 3  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
2 pntlem1.a . . 3  |-  ( ph  ->  A  e.  RR+ )
3 pntlem1.b . . 3  |-  ( ph  ->  B  e.  RR+ )
4 pntlem1.l . . 3  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
5 pntlem1.d . . 3  |-  D  =  ( A  +  1 )
6 pntlem1.f . . 3  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
7 pntlem1.u . . 3  |-  ( ph  ->  U  e.  RR+ )
8 pntlem1.u2 . . 3  |-  ( ph  ->  U  <_  A )
9 pntlem1.e . . 3  |-  E  =  ( U  /  D
)
10 pntlem1.k . . 3  |-  K  =  ( exp `  ( B  /  E ) )
11 pntlem1.y . . 3  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
12 pntlem1.x . . 3  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
13 pntlem1.c . . 3  |-  ( ph  ->  C  e.  RR+ )
14 pntlem1.w . . 3  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14pntlema 21295 . 2  |-  ( ph  ->  W  e.  RR+ )
162adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  A  e.  RR+ )
173adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  B  e.  RR+ )
184adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  L  e.  ( 0 (,) 1
) )
197adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  U  e.  RR+ )
208adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  U  <_  A )
2111adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( Y  e.  RR+  /\  1  <_  Y ) )
2212adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( X  e.  RR+  /\  Y  < 
X ) )
2313adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  C  e.  RR+ )
24 simpr 449 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  v  e.  ( W [,)  +oo )
)
25 eqid 2438 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  1 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 )
26 eqid 2438 . . . 4  |-  ( |_
`  ( ( ( log `  v )  /  ( log `  K
) )  /  2
) )  =  ( |_ `  ( ( ( log `  v
)  /  ( log `  K ) )  / 
2 ) )
27 pntleme.U . . . . 5  |-  ( ph  ->  A. z  e.  ( Y [,)  +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
2827adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  A. z  e.  ( Y [,)  +oo ) ( abs `  (
( R `  z
)  /  z ) )  <_  U )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 21294 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
3029simp2d 971 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
3130rpxrd 10654 . . . . . . 7  |-  ( ph  ->  K  e.  RR* )
32 pnfxr 10718 . . . . . . . 8  |-  +oo  e.  RR*
3332a1i 11 . . . . . . 7  |-  ( ph  ->  +oo  e.  RR* )
3430rpred 10653 . . . . . . . 8  |-  ( ph  ->  K  e.  RR )
35 ltpnf 10726 . . . . . . . 8  |-  ( K  e.  RR  ->  K  <  +oo )
3634, 35syl 16 . . . . . . 7  |-  ( ph  ->  K  <  +oo )
37 lbico1 10971 . . . . . . 7  |-  ( ( K  e.  RR*  /\  +oo  e.  RR*  /\  K  <  +oo )  ->  K  e.  ( K [,)  +oo ) )
3831, 33, 36, 37syl3anc 1185 . . . . . 6  |-  ( ph  ->  K  e.  ( K [,)  +oo ) )
39 pntleme.K . . . . . 6  |-  ( ph  ->  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  <  z  /\  (
( 1  +  ( L  x.  E ) )  x.  z )  <  ( k  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
40 oveq1 6091 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
k  x.  y )  =  ( K  x.  y ) )
4140breq2d 4227 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( k  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) ) )
4241anbi2d 686 . . . . . . . . . 10  |-  ( k  =  K  ->  (
( y  <  z  /\  ( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( k  x.  y ) )  <->  ( y  <  z  /\  ( ( 1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) ) ) )
4342anbi1d 687 . . . . . . . . 9  |-  ( k  =  K  ->  (
( ( y  < 
z  /\  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( k  x.  y
) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E
) )  x.  z
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
)  <->  ( ( y  <  z  /\  (
( 1  +  ( L  x.  E ) )  x.  z )  <  ( K  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )
4443rexbidv 2728 . . . . . . . 8  |-  ( k  =  K  ->  ( E. z  e.  RR+  (
( y  <  z  /\  ( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( k  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E )  <->  E. z  e.  RR+  ( ( y  <  z  /\  (
( 1  +  ( L  x.  E ) )  x.  z )  <  ( K  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )
4544ralbidv 2727 . . . . . . 7  |-  ( k  =  K  ->  ( A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  <  z  /\  (
( 1  +  ( L  x.  E ) )  x.  z )  <  ( k  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E )  <->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
z  /\  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E
) )  x.  z
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) ) )
4645rspcva 3052 . . . . . 6  |-  ( ( K  e.  ( K [,)  +oo )  /\  A. k  e.  ( K [,)  +oo ) A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
z  /\  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( k  x.  y
) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E
) )  x.  z
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) )  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
z  /\  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E
) )  x.  z
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) )
4738, 39, 46syl2anc 644 . . . . 5  |-  ( ph  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  (
( y  <  z  /\  ( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
4847adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  A. y  e.  ( X (,)  +oo ) E. z  e.  RR+  ( ( y  < 
z  /\  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E
) )  x.  z
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) )
49 pntleme.C . . . . 5  |-  ( ph  ->  A. z  e.  ( 1 (,)  +oo )
( ( ( ( abs `  ( R `
 z ) )  x.  ( log `  z
) )  -  (
( 2  /  ( log `  z ) )  x.  sum_ i  e.  ( 1 ... ( |_
`  ( z  /  Y ) ) ) ( ( abs `  ( R `  ( z  /  i ) ) )  x.  ( log `  i ) ) ) )  /  z )  <_  C )
5049adantr 453 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  A. z  e.  ( 1 (,)  +oo ) ( ( ( ( abs `  ( R `  z )
)  x.  ( log `  z ) )  -  ( ( 2  / 
( log `  z
) )  x.  sum_ i  e.  ( 1 ... ( |_ `  ( z  /  Y
) ) ) ( ( abs `  ( R `  ( z  /  i ) ) )  x.  ( log `  i ) ) ) )  /  z )  <_  C )
511, 16, 17, 18, 5, 6, 19, 20, 9, 10, 21, 22, 23, 14, 24, 25, 26, 28, 48, 50pntlemo 21306 . . 3  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
5251ralrimiva 2791 . 2  |-  ( ph  ->  A. v  e.  ( W [,)  +oo )
( abs `  (
( R `  v
)  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
53 oveq1 6091 . . . 4  |-  ( w  =  W  ->  (
w [,)  +oo )  =  ( W [,)  +oo ) )
5453raleqdv 2912 . . 3  |-  ( w  =  W  ->  ( A. v  e.  (
w [,)  +oo ) ( abs `  ( ( R `  v )  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) )  <->  A. v  e.  ( W [,)  +oo )
( abs `  (
( R `  v
)  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) ) )
5554rspcev 3054 . 2  |-  ( ( W  e.  RR+  /\  A. v  e.  ( W [,)  +oo ) ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
5615, 52, 55syl2anc 644 1  |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,)  +oo ) ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    +oocpnf 9122   RR*cxr 9124    < clt 9125    <_ cle 9126    - cmin 9296    / cdiv 9682   2c2 10054   3c3 10055   4c4 10056  ;cdc 10387   RR+crp 10617   (,)cioo 10921   [,)cico 10923   [,]cicc 10924   ...cfz 11048   |_cfl 11206   ^cexp 11387   abscabs 12044   sum_csu 12484   expce 12669   logclog 20457  ψcchp 20880
This theorem is referenced by:  pntlemp  21309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-e 12676  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-em 20836  df-vma 20885  df-chp 20886
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