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Theorem pntlemi 21251
Description: Lemma for pnt 21261. Eliminate some assumptions from pntlemj 21250. (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 21242 . . . . . . 7  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1211simp2d 970 . . . . . 6  |-  ( ph  ->  K  e.  RR+ )
13 elfzoelz 11095 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
14 rpexpcl 11355 . . . . . 6  |-  ( ( K  e.  RR+  /\  J  e.  ZZ )  ->  ( K ^ J )  e.  RR+ )
1512, 13, 14syl2an 464 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR+ )
1615rpred 10604 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR )
17 elfzofz 11109 . . . . . 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 21246 . . . . . 6  |-  ( (
ph  /\  J  e.  ( M ... N ) )  ->  ( X  <  ( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2617, 25sylan2 461 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( X  < 
( K ^ J
)  /\  ( K ^ J )  <_  ( sqr `  Z ) ) )
2726simpld 446 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  <  ( K ^ J ) )
2819simpld 446 . . . . . 6  |-  ( ph  ->  X  e.  RR+ )
2928adantr 452 . . . . 5  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  X  e.  RR+ )
30 rpxr 10575 . . . . 5  |-  ( X  e.  RR+  ->  X  e. 
RR* )
31 elioopnf 10954 . . . . 5  |-  ( X  e.  RR*  ->  ( ( K ^ J )  e.  ( X (,)  +oo )  <->  ( ( K ^ J )  e.  RR  /\  X  < 
( K ^ J
) ) ) )
3229, 30, 313syl 19 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( ( K ^ J )  e.  ( X (,)  +oo ) 
<->  ( ( K ^ J )  e.  RR  /\  X  <  ( K ^ J ) ) ) )
3316, 27, 32mpbir2and 889 . . 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 452 . . 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 4176 . . . . . . . 8  |-  ( z  =  x  ->  (
y  <  z  <->  y  <  x ) )
37 oveq2 6048 . . . . . . . . 9  |-  ( z  =  x  ->  (
( 1  +  ( L  x.  E ) )  x.  z )  =  ( ( 1  +  ( L  x.  E ) )  x.  x ) )
3837breq1d 4182 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y )  <->  ( (
1  +  ( L  x.  E ) )  x.  x )  < 
( K  x.  y
) ) )
3936, 38anbi12d 692 . . . . . . 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 20 . . . . . . . . 9  |-  ( z  =  x  ->  z  =  x )
4140, 37oveq12d 6058 . . . . . . . 8  |-  ( z  =  x  ->  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) )  =  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) )
4241raleqdv 2870 . . . . . . 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 692 . . . . . 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 2893 . . . . 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 4175 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
y  <  x  <->  ( K ^ J )  <  x
) )
46 oveq2 6048 . . . . . . . . 9  |-  ( y  =  ( K ^ J )  ->  ( K  x.  y )  =  ( K  x.  ( K ^ J ) ) )
4746breq2d 4184 . . . . . . . 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 692 . . . . . . 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 686 . . . . . 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 2687 . . . . 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 249 . . . 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 3008 . . 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 58 . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 707 . . 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 733 . . 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 734 . . 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 732 . . 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 2404 . . 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 21250 . 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 2794 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   3c3 10006   4c4 10007   8c8 10011   ZZcz 10238  ;cdc 10338   RR+crp 10568   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   ...cfz 10999  ..^cfzo 11090   |_cfl 11156   ^cexp 11337   sqrcsqr 11993   abscabs 11994   sum_csu 12434   expce 12619   logclog 20405  ψcchp 20828
This theorem is referenced by:  pntlemf  21252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-vma 20833  df-chp 20834
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