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Theorem pntlemi 23929
Description: Lemma for pnt 23939. Eliminate some assumptions from pntlemj 23928. (Contributed by Mario Carneiro, 13-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
) ) ) ) )
pntlem1.z  |-  ( ph  ->  Z  e.  ( W [,) +oo ) )
pntlem1.m  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
pntlem1.n  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
pntlem1.U  |-  ( ph  ->  A. z  e.  ( Y [,) +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
pntlem1.K  |-  ( 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 ) )
pntlem1.o  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
Assertion
Ref Expression
pntlemi  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
Distinct variable groups:    z, C    y, n, z, J    u, n, L, y, z    n, K, y, z    n, M, z    n, O, z    ph, n    n, N, z    R, n, u, y, z    U, n, z    n, W, z    n, X, y, z    n, Y, z   
n, a, u, y, z, E    n, Z, u, z
Allowed substitution hints:    ph( y, z, u, a)    A( y, z, u, n, a)    B( y, z, u, n, a)    C( y, u, n, a)    D( y, z, u, n, a)    R( a)    U( y, u, a)    F( y, z, u, n, a)    J( u, a)    K( u, a)    L( a)    M( y, u, a)    N( y, u, a)    O( y, u, a)    W( y, u, a)    X( u, a)    Y( y, u, a)    Z( y, a)

Proof of Theorem pntlemi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pntlem1.r . . . . . . . 8  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
2 pntlem1.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
3 pntlem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR+ )
4 pntlem1.l . . . . . . . 8  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
5 pntlem1.d . . . . . . . 8  |-  D  =  ( A  +  1 )
6 pntlem1.f . . . . . . . 8  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
7 pntlem1.u . . . . . . . 8  |-  ( ph  ->  U  e.  RR+ )
8 pntlem1.u2 . . . . . . . 8  |-  ( ph  ->  U  <_  A )
9 pntlem1.e . . . . . . . 8  |-  E  =  ( U  /  D
)
10 pntlem1.k . . . . . . . 8  |-  K  =  ( exp `  ( B  /  E ) )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 23920 . . . . . . 7  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1211simp2d 1007 . . . . . 6  |-  ( ph  ->  K  e.  RR+ )
13 elfzoelz 11744 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
14 rpexpcl 12111 . . . . . 6  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( K ^ J )  e.  RR+ )
1512, 13, 14syl2an 475 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR+ )
1615rpred 11199 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR )
17 elfzofz 11759 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ( M ... N ) )
18 pntlem1.y . . . . . . 7  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
19 pntlem1.x . . . . . . 7  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
20 pntlem1.c . . . . . . 7  |-  ( ph  ->  C  e.  RR+ )
21 pntlem1.w . . . . . . 7  |-  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
) ) ) ) )
22 pntlem1.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( W [,) +oo ) )
23 pntlem1.m . . . . . . 7  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
24 pntlem1.n . . . . . . 7  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
251, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 19, 20, 21, 22, 23, 24pntlemh 23924 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2617, 25sylan2 472 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( X  < 
( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2726simpld 457 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  <  ( K ^ J ) )
2819simpld 457 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
2928adantr 463 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  e.  RR+ )
30 rpxr 11168 . . . . 5  |-  ( X  e.  RR+  ->  X  e. 
RR* )
31 elioopnf 11561 . . . . 5  |-  ( X  e.  RR*  ->  ( ( K ^ J )  e.  ( X (,) +oo )  <->  ( ( K ^ J )  e.  RR  /\  X  < 
( K ^ J
) ) ) )
3229, 30, 313syl 20 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( ( K ^ J )  e.  ( X (,) +oo ) 
<->  ( ( K ^ J )  e.  RR  /\  X  <  ( K ^ J ) ) ) )
3316, 27, 32mpbir2and 920 . . 3  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  ( X (,) +oo )
)
34 pntlem1.K . . . 4  |-  ( 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 ) )
3534adantr 463 . . 3  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  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 ) )
36 breq2 4388 . . . . . . . 8  |-  ( z  =  x  ->  (
y  <  z  <->  y  <  x ) )
37 oveq2 6226 . . . . . . . . 9  |-  ( z  =  x  ->  (
( 1  +  ( L  x.  E ) )  x.  z )  =  ( ( 1  +  ( L  x.  E ) )  x.  x ) )
3837breq1d 4394 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  y
) ) )
3936, 38anbi12d 708 . . . . . . 7  |-  ( z  =  x  ->  (
( y  <  z  /\  ( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y ) )  <->  ( y  <  x  /\  ( ( 1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  y
) ) ) )
40 id 22 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
4140, 37oveq12d 6236 . . . . . . . 8  |-  ( z  =  x  ->  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) )  =  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) )
4241raleqdv 3002 . . . . . . 7  |-  ( z  =  x  ->  ( A. u  e.  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  (
( R `  u
)  /  u ) )  <_  E  <->  A. u  e.  ( x [,] (
( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  (
( R `  u
)  /  u ) )  <_  E )
)
4339, 42anbi12d 708 . . . . . 6  |-  ( z  =  x  ->  (
( ( 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  <  x  /\  (
( 1  +  ( L  x.  E ) )  x.  x )  <  ( K  x.  y ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )
4443cbvrexv 3027 . . . . 5  |-  ( 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. x  e.  RR+  ( ( y  <  x  /\  (
( 1  +  ( L  x.  E ) )  x.  x )  <  ( K  x.  y ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
45 breq1 4387 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
y  <  x  <->  ( K ^ J )  <  x
) )
46 oveq2 6226 . . . . . . . . 9  |-  ( y  =  ( K ^ J )  ->  ( K  x.  y )  =  ( K  x.  ( K ^ J ) ) )
4746breq2d 4396 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  ( K ^ J ) ) ) )
4845, 47anbi12d 708 . . . . . . 7  |-  ( y  =  ( K ^ J )  ->  (
( y  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  y ) )  <->  ( ( K ^ J )  < 
x  /\  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  ( K ^ J ) ) ) ) )
4948anbi1d 702 . . . . . 6  |-  ( y  =  ( K ^ J )  ->  (
( ( y  < 
x  /\  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  y
) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
)  <->  ( ( ( K ^ J )  <  x  /\  (
( 1  +  ( L  x.  E ) )  x.  x )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) ) )
5049rexbidv 2910 . . . . 5  |-  ( y  =  ( K ^ J )  ->  ( E. x  e.  RR+  (
( y  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  y ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E )  <->  E. x  e.  RR+  ( ( ( K ^ J )  <  x  /\  (
( 1  +  ( L  x.  E ) )  x.  x )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) ) )
5144, 50syl5bb 257 . . . 4  |-  ( y  =  ( K ^ J )  ->  ( 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. x  e.  RR+  ( ( ( K ^ J )  <  x  /\  (
( 1  +  ( L  x.  E ) )  x.  x )  <  ( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) ( abs `  ( ( R `  u )  /  u
) )  <_  E
) ) )
5251rspcv 3148 . . 3  |-  ( ( K ^ J )  e.  ( X (,) +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 )  ->  E. x  e.  RR+  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )
5333, 35, 52sylc 60 . 2  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  E. x  e.  RR+  ( ( ( K ^ J )  < 
x  /\  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( x [,] (
( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  (
( R `  u
)  /  u ) )  <_  E )
)
542ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  A  e.  RR+ )
553ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  B  e.  RR+ )
564ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  L  e.  ( 0 (,) 1
) )
577ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  U  e.  RR+ )
588ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  U  <_  A )
5918ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  ( Y  e.  RR+  /\  1  <_  Y ) )
6019ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  ( X  e.  RR+  /\  Y  < 
X ) )
6120ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  C  e.  RR+ )
6222ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  Z  e.  ( W [,) +oo )
)
63 pntlem1.U . . . 4  |-  ( ph  ->  A. z  e.  ( Y [,) +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
6463ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  A. z  e.  ( Y [,) +oo ) ( abs `  (
( R `  z
)  /  z ) )  <_  U )
6534ad2antrr 723 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( 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
) )
66 pntlem1.o . . 3  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
67 simprl 754 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  x  e.  RR+ )
68 simprr 755 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  ( (
( K ^ J
)  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
69 simplr 753 . . 3  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  J  e.  ( M..^ N ) )
70 eqid 2396 . . 3  |-  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  x ) ) )  +  1 ) ... ( |_ `  ( Z  /  x ) ) )  =  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  x ) ) )  +  1 ) ... ( |_ `  ( Z  /  x ) ) )
711, 54, 55, 56, 5, 6, 57, 58, 9, 10, 59, 60, 61, 21, 62, 23, 24, 64, 65, 66, 67, 68, 69, 70pntlemj 23928 . 2  |-  ( ( ( ph  /\  J  e.  ( M..^ N ) )  /\  ( x  e.  RR+  /\  (
( ( K ^ J )  <  x  /\  ( ( 1  +  ( L  x.  E
) )  x.  x
)  <  ( K  x.  ( K ^ J
) ) )  /\  A. u  e.  ( x [,] ( ( 1  +  ( L  x.  E ) )  x.  x ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
7253, 71rexlimddv 2892 1  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( ( U  -  E )  x.  ( ( ( L  x.  E )  / 
8 )  x.  ( log `  Z ) ) )  <_  sum_ n  e.  O  ( ( ( U  /  n )  -  ( abs `  (
( R `  ( Z  /  n ) )  /  Z ) ) )  x.  ( log `  n ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2746   E.wrex 2747   class class class wbr 4384    |-> cmpt 4442   ` cfv 5513  (class class class)co 6218   RRcr 9424   0cc0 9425   1c1 9426    + caddc 9428    x. cmul 9430   +oocpnf 9558   RR*cxr 9560    < clt 9561    <_ cle 9562    - cmin 9740    / cdiv 10145   2c2 10524   3c3 10525   4c4 10526   8c8 10530   ZZcz 10803  ;cdc 10917   RR+crp 11161   (,)cioo 11472   [,)cico 11474   [,]cicc 11475   ...cfz 11615  ..^cfzo 11739   |_cfl 11849   ^cexp 12092   sqrcsqrt 13091   abscabs 13092   sum_csu 13533   expce 13822   logclog 23050  ψcchp 23506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-inf2 7994  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503  ax-addf 9504  ax-mulf 9505
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-iin 4263  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-se 4770  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-of 6461  df-om 6622  df-1st 6721  df-2nd 6722  df-supp 6840  df-recs 6982  df-rdg 7016  df-1o 7070  df-2o 7071  df-oadd 7074  df-er 7251  df-map 7362  df-pm 7363  df-ixp 7411  df-en 7458  df-dom 7459  df-sdom 7460  df-fin 7461  df-fsupp 7767  df-fi 7808  df-sup 7838  df-oi 7872  df-card 8255  df-cda 8483  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-4 10535  df-5 10536  df-6 10537  df-7 10538  df-8 10539  df-9 10540  df-10 10541  df-n0 10735  df-z 10804  df-dec 10918  df-uz 11024  df-q 11124  df-rp 11162  df-xneg 11261  df-xadd 11262  df-xmul 11263  df-ioo 11476  df-ioc 11477  df-ico 11478  df-icc 11479  df-fz 11616  df-fzo 11740  df-fl 11851  df-mod 11920  df-seq 12034  df-exp 12093  df-fac 12279  df-bc 12306  df-hash 12331  df-shft 12925  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-limsup 13319  df-clim 13336  df-rlim 13337  df-sum 13534  df-ef 13828  df-e 13829  df-sin 13830  df-cos 13831  df-pi 13833  df-dvds 14012  df-gcd 14170  df-prm 14243  df-pc 14386  df-struct 14659  df-ndx 14660  df-slot 14661  df-base 14662  df-sets 14663  df-ress 14664  df-plusg 14738  df-mulr 14739  df-starv 14740  df-sca 14741  df-vsca 14742  df-ip 14743  df-tset 14744  df-ple 14745  df-ds 14747  df-unif 14748  df-hom 14749  df-cco 14750  df-rest 14853  df-topn 14854  df-0g 14872  df-gsum 14873  df-topgen 14874  df-pt 14875  df-prds 14878  df-xrs 14932  df-qtop 14937  df-imas 14938  df-xps 14940  df-mre 15016  df-mrc 15017  df-acs 15019  df-mgm 16012  df-sgrp 16051  df-mnd 16061  df-submnd 16107  df-mulg 16200  df-cntz 16495  df-cmn 16940  df-psmet 18547  df-xmet 18548  df-met 18549  df-bl 18550  df-mopn 18551  df-fbas 18552  df-fg 18553  df-cnfld 18557  df-top 19507  df-bases 19509  df-topon 19510  df-topsp 19511  df-cld 19628  df-ntr 19629  df-cls 19630  df-nei 19708  df-lp 19746  df-perf 19747  df-cn 19837  df-cnp 19838  df-haus 19925  df-tx 20171  df-hmeo 20364  df-fil 20455  df-fm 20547  df-flim 20548  df-flf 20549  df-xms 20931  df-ms 20932  df-tms 20933  df-cncf 21490  df-limc 22378  df-dv 22379  df-log 23052  df-vma 23511  df-chp 23512
This theorem is referenced by:  pntlemf  23930
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