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Theorem pntleme 20980
Description: Lemma for pnt 20986. Package up pntlemo 20979 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 20968 . 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 2366 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  1 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 )
26 eqid 2366 . . . 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 20967 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
3029simp2d 969 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
3130rpxrd 10542 . . . . . . 7  |-  ( ph  ->  K  e.  RR* )
32 pnfxr 10606 . . . . . . . 8  |-  +oo  e.  RR*
3332a1i 10 . . . . . . 7  |-  ( ph  ->  +oo  e.  RR* )
3430rpred 10541 . . . . . . . 8  |-  ( ph  ->  K  e.  RR )
35 ltpnf 10614 . . . . . . . 8  |-  ( K  e.  RR  ->  K  <  +oo )
3634, 35syl 15 . . . . . . 7  |-  ( ph  ->  K  <  +oo )
37 lbico1 10859 . . . . . . 7  |-  ( ( K  e.  RR*  /\  +oo  e.  RR*  /\  K  <  +oo )  ->  K  e.  ( K [,)  +oo ) )
3831, 33, 36, 37syl3anc 1183 . . . . . 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 5988 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
k  x.  y )  =  ( K  x.  y ) )
4140breq2d 4137 . . . . . . . . . . 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 2649 . . . . . . . 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 2648 . . . . . . 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 2967 . . . . . 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 20979 . . 3  |-  ( (
ph  /\  v  e.  ( W [,)  +oo )
)  ->  ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
5251ralrimiva 2711 . 2  |-  ( ph  ->  A. v  e.  ( W [,)  +oo )
( abs `  (
( R `  v
)  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
53 oveq1 5988 . . . 4  |-  ( w  =  W  ->  (
w [,)  +oo )  =  ( W [,)  +oo ) )
5453raleqdv 2827 . . 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 2969 . 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 935    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   class class class wbr 4125    e. cmpt 4179   ` cfv 5358  (class class class)co 5981   RRcr 8883   0cc0 8884   1c1 8885    + caddc 8887    x. cmul 8889    +oocpnf 9011   RR*cxr 9013    < clt 9014    <_ cle 9015    - cmin 9184    / cdiv 9570   2c2 9942   3c3 9943   4c4 9944  ;cdc 10275   RR+crp 10505   (,)cioo 10809   [,)cico 10811   [,]cicc 10812   ...cfz 10935   |_cfl 11088   ^cexp 11269   abscabs 11926   sum_csu 12366   expce 12551   logclog 20130  ψcchp 20553
This theorem is referenced by:  pntlemp  20982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ioc 10814  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557  df-e 12558  df-sin 12559  df-cos 12560  df-pi 12562  df-dvds 12740  df-gcd 12894  df-prm 12967  df-pc 13098  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432  df-log 20132  df-em 20509  df-vma 20558  df-chp 20559
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