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Theorem pntleme 20757
Description: Lemma for pnt 20763. Package up pntlemo 20756 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 20745 . 2  |-  ( ph  ->  W  e.  RR+ )
162adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  A  e.  RR+ )
173adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  B  e.  RR+ )
184adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  L  e.  ( 0 (,) 1
) )
197adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  U  e.  RR+ )
208adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  U  <_  A )
2111adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( Y  e.  RR+  /\  1  <_  Y ) )
2212adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( X  e.  RR+  /\  Y  < 
X ) )
2313adantr 451 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  C  e.  RR+ )
24 simpr 447 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  v  e.  ( W [,)  +oo )
)
25 eqid 2283 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  1 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 )
26 eqid 2283 . . . 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 451 . . . 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 20744 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
3029simp2d 968 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
3130rpxrd 10391 . . . . . . 7  |-  ( ph  ->  K  e.  RR* )
32 pnfxr 10455 . . . . . . . 8  |-  +oo  e.  RR*
3332a1i 10 . . . . . . 7  |-  ( ph  ->  +oo  e.  RR* )
3430rpred 10390 . . . . . . . 8  |-  ( ph  ->  K  e.  RR )
35 ltpnf 10463 . . . . . . . 8  |-  ( K  e.  RR  ->  K  <  +oo )
3634, 35syl 15 . . . . . . 7  |-  ( ph  ->  K  <  +oo )
37 lbico1 10706 . . . . . . 7  |-  ( ( K  e.  RR*  /\  +oo  e.  RR*  /\  K  <  +oo )  ->  K  e.  ( K [,)  +oo ) )
3831, 33, 36, 37syl3anc 1182 . . . . . 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 5865 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
k  x.  y )  =  ( K  x.  y ) )
4140breq2d 4035 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( k  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) ) )
4241anbi2d 684 . . . . . . . . . 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 685 . . . . . . . . 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 2564 . . . . . . . 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 2563 . . . . . . 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 2882 . . . . . 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 642 . . . . 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 451 . . . 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 451 . . . 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 20756 . . 3  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
5251ralrimiva 2626 . 2  |-  ( ph  ->  A. v  e.  ( W [,)  +oo )
( abs `  (
( R `  v
)  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
53 oveq1 5865 . . . 4  |-  ( w  =  W  ->  (
w [,)  +oo )  =  ( W [,)  +oo ) )
5453raleqdv 2742 . . 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 2884 . 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 642 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797  ;cdc 10124   RR+crp 10354   (,)cioo 10656   [,)cico 10658   [,]cicc 10659   ...cfz 10782   |_cfl 10924   ^cexp 11104   abscabs 11719   sum_csu 12158   expce 12343   logclog 19912  ψcchp 20330
This theorem is referenced by:  pntlemp  20759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-em 20287  df-vma 20335  df-chp 20336
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