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Theorem pntlemi 23510
Description: Lemma for pnt 23520. Eliminate some assumptions from pntlemj 23509. (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 23501 . . . . . . 7  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1211simp2d 1004 . . . . . 6  |-  ( ph  ->  K  e.  RR+ )
13 elfzoelz 11786 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
14 rpexpcl 12141 . . . . . 6  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( K ^ J )  e.  RR+ )
1512, 13, 14syl2an 477 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR+ )
1615rpred 11245 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR )
17 elfzofz 11800 . . . . . 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 23505 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2617, 25sylan2 474 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( X  < 
( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2726simpld 459 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  <  ( K ^ J ) )
2819simpld 459 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
2928adantr 465 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  e.  RR+ )
30 rpxr 11216 . . . . 5  |-  ( X  e.  RR+  ->  X  e. 
RR* )
31 elioopnf 11607 . . . . 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 915 . . 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 465 . . 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 4444 . . . . . . . 8  |-  ( z  =  x  ->  (
y  <  z  <->  y  <  x ) )
37 oveq2 6283 . . . . . . . . 9  |-  ( z  =  x  ->  (
( 1  +  ( L  x.  E ) )  x.  z )  =  ( ( 1  +  ( L  x.  E ) )  x.  x ) )
3837breq1d 4450 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  y
) ) )
3936, 38anbi12d 710 . . . . . . 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 6293 . . . . . . . 8  |-  ( z  =  x  ->  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) )  =  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) )
4241raleqdv 3057 . . . . . . 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 710 . . . . . 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 3082 . . . . 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 4443 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
y  <  x  <->  ( K ^ J )  <  x
) )
46 oveq2 6283 . . . . . . . . 9  |-  ( y  =  ( K ^ J )  ->  ( K  x.  y )  =  ( K  x.  ( K ^ J ) ) )
4746breq2d 4452 . . . . . . . 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 710 . . . . . . 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 704 . . . . . 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 2966 . . . . 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 3203 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 725 . . 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 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 ) ) )  ->  x  e.  RR+ )
68 simprr 756 . . 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 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 ) ) )  ->  J  e.  ( M..^ N ) )
70 eqid 2460 . . 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 23509 . 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 2952 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   class class class wbr 4440    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794    / cdiv 10195   2c2 10574   3c3 10575   4c4 10576   8c8 10580   ZZcz 10853  ;cdc 10965   RR+crp 11209   (,)cioo 11518   [,)cico 11520   [,]cicc 11521   ...cfz 11661  ..^cfzo 11781   |_cfl 11884   ^cexp 12122   sqrcsqr 13016   abscabs 13017   sum_csu 13457   expce 13648   logclog 22663  ψcchp 23087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-e 13655  df-sin 13656  df-cos 13657  df-pi 13659  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-log 22665  df-vma 23092  df-chp 23093
This theorem is referenced by:  pntlemf  23511
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