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Theorem pntlemi 22833
Description: Lemma for pnt 22843. Eliminate some assumptions from pntlemj 22832. (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 22824 . . . . . . 7  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1211simp2d 1001 . . . . . 6  |-  ( ph  ->  K  e.  RR+ )
13 elfzoelz 11545 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
14 rpexpcl 11876 . . . . . 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 11019 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR )
17 elfzofz 11559 . . . . . 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 22828 . . . . . 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 10990 . . . . 5  |-  ( X  e.  RR+  ->  X  e. 
RR* )
31 elioopnf 11375 . . . . 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 913 . . 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 4291 . . . . . . . 8  |-  ( z  =  x  ->  (
y  <  z  <->  y  <  x ) )
37 oveq2 6094 . . . . . . . . 9  |-  ( z  =  x  ->  (
( 1  +  ( L  x.  E ) )  x.  z )  =  ( ( 1  +  ( L  x.  E ) )  x.  x ) )
3837breq1d 4297 . . . . . . . 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 6104 . . . . . . . 8  |-  ( z  =  x  ->  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) )  =  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) )
4241raleqdv 2918 . . . . . . 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 2943 . . . . 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 4290 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
y  <  x  <->  ( K ^ J )  <  x
) )
46 oveq2 6094 . . . . . . . . 9  |-  ( y  =  ( K ^ J )  ->  ( K  x.  y )  =  ( K  x.  ( K ^ J ) ) )
4746breq2d 4299 . . . . . . . 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 2731 . . . . 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 3064 . . 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 2438 . . 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 22832 . 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 2840 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 965    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   +oocpnf 9407   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   2c2 10363   3c3 10364   4c4 10365   8c8 10369   ZZcz 10638  ;cdc 10747   RR+crp 10983   (,)cioo 11292   [,)cico 11294   [,]cicc 11295   ...cfz 11429  ..^cfzo 11540   |_cfl 11632   ^cexp 11857   sqrcsqr 12714   abscabs 12715   sum_csu 13155   expce 13339   logclog 21986  ψcchp 22410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-vma 22415  df-chp 22416
This theorem is referenced by:  pntlemf  22834
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