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Theorem pntleme 24458
Description: Lemma for pnt 24464. Package up pntlemo 24457 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 24446 . 2  |-  ( ph  ->  W  e.  RR+ )
162adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  A  e.  RR+ )
173adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  B  e.  RR+ )
184adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  L  e.  ( 0 (,) 1
) )
197adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  U  e.  RR+ )
208adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  U  <_  A )
2111adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  ( Y  e.  RR+  /\  1  <_  Y ) )
2212adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  ( X  e.  RR+  /\  Y  < 
X ) )
2313adantr 467 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  C  e.  RR+ )
24 simpr 463 . . . 4  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  v  e.  ( W [,) +oo )
)
25 eqid 2453 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  1 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  1 )
26 eqid 2453 . . . 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 467 . . . 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 24445 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
3029simp2d 1022 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
3130rpxrd 11349 . . . . . . 7  |-  ( ph  ->  K  e.  RR* )
32 pnfxr 11419 . . . . . . . 8  |- +oo  e.  RR*
3332a1i 11 . . . . . . 7  |-  ( ph  -> +oo  e.  RR* )
3430rpred 11348 . . . . . . . 8  |-  ( ph  ->  K  e.  RR )
35 ltpnf 11429 . . . . . . . 8  |-  ( K  e.  RR  ->  K  < +oo )
3634, 35syl 17 . . . . . . 7  |-  ( ph  ->  K  < +oo )
37 lbico1 11696 . . . . . . 7  |-  ( ( K  e.  RR*  /\ +oo  e.  RR*  /\  K  < +oo )  ->  K  e.  ( K [,) +oo ) )
3831, 33, 36, 37syl3anc 1269 . . . . . 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 6302 . . . . . . . . . . . 12  |-  ( k  =  K  ->  (
k  x.  y )  =  ( K  x.  y ) )
4140breq2d 4417 . . . . . . . . . . 11  |-  ( k  =  K  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( k  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  z )  < 
( K  x.  y
) ) )
4241anbi2d 711 . . . . . . . . . 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 712 . . . . . . . . 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 2903 . . . . . . . 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 2829 . . . . . . 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 3150 . . . . . 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 667 . . . . 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 467 . . . 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 467 . . . 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 24457 . . 3  |-  ( (
ph  /\  v  e.  ( W [,) +oo )
)  ->  ( abs `  ( ( R `  v )  /  v
) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
5251ralrimiva 2804 . 2  |-  ( ph  ->  A. v  e.  ( W [,) +oo )
( abs `  (
( R `  v
)  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
53 oveq1 6302 . . . 4  |-  ( w  =  W  ->  (
w [,) +oo )  =  ( W [,) +oo ) )
5453raleqdv 2995 . . 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 3152 . 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 667 1  |-  ( ph  ->  E. w  e.  RR+  A. v  e.  ( w [,) +oo ) ( abs `  ( ( R `  v )  /  v ) )  <_  ( U  -  ( F  x.  ( U ^ 3 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   class class class wbr 4405    |-> cmpt 4464   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   1c1 9545    + caddc 9547    x. cmul 9549   +oocpnf 9677   RR*cxr 9679    < clt 9680    <_ cle 9681    - cmin 9865    / cdiv 10276   2c2 10666   3c3 10667   4c4 10668  ;cdc 11058   RR+crp 11309   (,)cioo 11642   [,)cico 11644   [,]cicc 11645   ...cfz 11791   |_cfl 12033   ^cexp 12279   abscabs 13309   sum_csu 13764   expce 14126   logclog 23516  ψcchp 24031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-shft 13142  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-limsup 13538  df-clim 13564  df-rlim 13565  df-sum 13765  df-ef 14133  df-e 14134  df-sin 14135  df-cos 14136  df-pi 14138  df-dvds 14318  df-gcd 14481  df-prm 14635  df-pc 14799  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-fbas 18979  df-fg 18980  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-lp 20164  df-perf 20165  df-cn 20255  df-cnp 20256  df-haus 20343  df-tx 20589  df-hmeo 20782  df-fil 20873  df-fm 20965  df-flim 20966  df-flf 20967  df-xms 21347  df-ms 21348  df-tms 21349  df-cncf 21922  df-limc 22833  df-dv 22834  df-log 23518  df-em 23930  df-vma 24036  df-chp 24037
This theorem is referenced by:  pntlemp  24460
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