MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemi Unicode version

Theorem pntlemi 21167
Description: Lemma for pnt 21177. Eliminate some assumptions from pntlemj 21166. (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 21158 . . . . . . 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 11072 . . . . . 6  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
14 rpexpcl 11329 . . . . . 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 10582 . . . 4  |-  ( (
ph  /\  J  e.  ( M..^ N ) )  ->  ( K ^ J )  e.  RR )
17 elfzofz 11086 . . . . . 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 21162 . . . . . 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 10553 . . . . 5  |-  ( X  e.  RR+  ->  X  e. 
RR* )
31 elioopnf 10932 . . . . 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 4159 . . . . . . . 8  |-  ( z  =  x  ->  (
y  <  z  <->  y  <  x ) )
37 oveq2 6030 . . . . . . . . 9  |-  ( z  =  x  ->  (
( 1  +  ( L  x.  E ) )  x.  z )  =  ( ( 1  +  ( L  x.  E ) )  x.  x ) )
3837breq1d 4165 . . . . . . . 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 6040 . . . . . . . 8  |-  ( z  =  x  ->  (
z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) )  =  ( x [,] ( ( 1  +  ( L  x.  E
) )  x.  x
) ) )
4241raleqdv 2855 . . . . . . 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 2878 . . . . 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 4158 . . . . . . . 8  |-  ( y  =  ( K ^ J )  ->  (
y  <  x  <->  ( K ^ J )  <  x
) )
46 oveq2 6030 . . . . . . . . 9  |-  ( y  =  ( K ^ J )  ->  ( K  x.  y )  =  ( K  x.  ( K ^ J ) ) )
4746breq2d 4167 . . . . . . . 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 2672 . . . . 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 2993 . . 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 2389 . . 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 21166 . 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 2779 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 1717   A.wral 2651   E.wrex 2652   class class class wbr 4155    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    +oocpnf 9052   RR*cxr 9054    < clt 9055    <_ cle 9056    - cmin 9225    / cdiv 9611   2c2 9983   3c3 9984   4c4 9985   8c8 9989   ZZcz 10216  ;cdc 10316   RR+crp 10546   (,)cioo 10850   [,)cico 10852   [,]cicc 10853   ...cfz 10977  ..^cfzo 11067   |_cfl 11130   ^cexp 11311   sqrcsqr 11967   abscabs 11968   sum_csu 12408   expce 12593   logclog 20321  ψcchp 20744
This theorem is referenced by:  pntlemf  21168
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-ioc 10855  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-fac 11496  df-bc 11523  df-hash 11548  df-shft 11811  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-limsup 12194  df-clim 12211  df-rlim 12212  df-sum 12409  df-ef 12599  df-e 12600  df-sin 12601  df-cos 12602  df-pi 12604  df-dvds 12782  df-gcd 12936  df-prm 13009  df-pc 13140  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-fbas 16625  df-fg 16626  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-lp 17125  df-perf 17126  df-cn 17215  df-cnp 17216  df-haus 17303  df-tx 17517  df-hmeo 17710  df-fil 17801  df-fm 17893  df-flim 17894  df-flf 17895  df-xms 18261  df-ms 18262  df-tms 18263  df-cncf 18781  df-limc 19622  df-dv 19623  df-log 20323  df-vma 20749  df-chp 20750
  Copyright terms: Public domain W3C validator