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Theorem prmreclem6 12968
Description: Lemma for prmrec 12969. If the series  F was convergent, there would be some  k such that the sum starting from  k  +  1 sums to less than  1  /  2; this is a sufficient hypothesis for prmreclem5 12967 to produce the contradictory bound  N  /  2  < 
( 2 ^ k
) sqr N, which is false for  N  =  2 ^ ( 2 k  +  2 ). (Contributed by Mario Carneiro, 6-Aug-2014.)
Hypothesis
Ref Expression
prmrec.1  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 ) )
Assertion
Ref Expression
prmreclem6  |-  -.  seq  1 (  +  ,  F )  e.  dom  ~~>
Distinct variable group:    n, F

Proof of Theorem prmreclem6
Dummy variables  j 
k  m  p  r  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1z 10053 . . . . . . . . . . 11  |-  1  e.  ZZ
32a1i 10 . . . . . . . . . 10  |-  (  T. 
->  1  e.  ZZ )
4 nnrecre 9782 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
1  /  n )  e.  RR )
54adantl 452 . . . . . . . . . . . . 13  |-  ( (  T.  /\  n  e.  NN )  ->  (
1  /  n )  e.  RR )
6 0re 8838 . . . . . . . . . . . . 13  |-  0  e.  RR
7 ifcl 3601 . . . . . . . . . . . . 13  |-  ( ( ( 1  /  n
)  e.  RR  /\  0  e.  RR )  ->  if ( n  e. 
Prime ,  ( 1  /  n ) ,  0 )  e.  RR )
85, 6, 7sylancl 643 . . . . . . . . . . . 12  |-  ( (  T.  /\  n  e.  NN )  ->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 )  e.  RR )
9 prmrec.1 . . . . . . . . . . . 12  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 ) )
108, 9fmptd 5684 . . . . . . . . . . 11  |-  (  T. 
->  F : NN --> RR )
11 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  j  e.  NN )  ->  ( F `  j
)  e.  RR )
1210, 11sylan 457 . . . . . . . . . 10  |-  ( (  T.  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
131, 3, 12serfre 11075 . . . . . . . . 9  |-  (  T. 
->  seq  1 (  +  ,  F ) : NN --> RR )
1413trud 1314 . . . . . . . 8  |-  seq  1
(  +  ,  F
) : NN --> RR
15 frn 5395 . . . . . . . 8  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  ran  seq  1
(  +  ,  F
)  C_  RR )
1614, 15ax-mp 8 . . . . . . 7  |-  ran  seq  1 (  +  ,  F )  C_  RR
1716a1i 10 . . . . . 6  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  ran  seq  1 (  +  ,  F )  C_  RR )
18 1nn 9757 . . . . . . . . 9  |-  1  e.  NN
1914fdmi 5394 . . . . . . . . 9  |-  dom  seq  1 (  +  ,  F )  =  NN
2018, 19eleqtrri 2356 . . . . . . . 8  |-  1  e.  dom  seq  1 (  +  ,  F )
21 ne0i 3461 . . . . . . . . 9  |-  ( 1  e.  dom  seq  1
(  +  ,  F
)  ->  dom  seq  1
(  +  ,  F
)  =/=  (/) )
22 dm0rn0 4895 . . . . . . . . . 10  |-  ( dom 
seq  1 (  +  ,  F )  =  (/) 
<->  ran  seq  1 (  +  ,  F )  =  (/) )
2322necon3bii 2478 . . . . . . . . 9  |-  ( dom 
seq  1 (  +  ,  F )  =/=  (/) 
<->  ran  seq  1 (  +  ,  F )  =/=  (/) )
2421, 23sylib 188 . . . . . . . 8  |-  ( 1  e.  dom  seq  1
(  +  ,  F
)  ->  ran  seq  1
(  +  ,  F
)  =/=  (/) )
2520, 24ax-mp 8 . . . . . . 7  |-  ran  seq  1 (  +  ,  F )  =/=  (/)
2625a1i 10 . . . . . 6  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  ran  seq  1 (  +  ,  F )  =/=  (/) )
272a1i 10 . . . . . . . . 9  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  -> 
1  e.  ZZ )
28 climdm 12028 . . . . . . . . . 10  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  <->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1 (  +  ,  F ) ) )
2928biimpi 186 . . . . . . . . 9  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1 (  +  ,  F ) ) )
3014a1i 10 . . . . . . . . . 10  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  seq  1 (  +  ,  F ) : NN --> RR )
31 ffvelrn 5663 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F ) : NN --> RR  /\  k  e.  NN )  ->  (  seq  1 (  +  ,  F ) `  k
)  e.  RR )
3230, 31sylan 457 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq  1
(  +  ,  F
) `  k )  e.  RR )
331, 27, 29, 32climrecl 12057 . . . . . . . 8  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  -> 
(  ~~>  `  seq  1
(  +  ,  F
) )  e.  RR )
34 simpr 447 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  NN )
3529adantr 451 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq  1 (  +  ,  F )  ~~>  (  ~~>  `  seq  1
(  +  ,  F
) ) )
36 eleq1 2343 . . . . . . . . . . . . . . 15  |-  ( n  =  j  ->  (
n  e.  Prime  <->  j  e.  Prime ) )
37 oveq2 5866 . . . . . . . . . . . . . . 15  |-  ( n  =  j  ->  (
1  /  n )  =  ( 1  / 
j ) )
38 eqidd 2284 . . . . . . . . . . . . . . 15  |-  ( n  =  j  ->  0  =  0 )
3936, 37, 38ifbieq12d 3587 . . . . . . . . . . . . . 14  |-  ( n  =  j  ->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
40 prmnn 12761 . . . . . . . . . . . . . . . . . . 19  |-  ( j  e.  Prime  ->  j  e.  NN )
4140adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( (  T.  /\  j  e. 
Prime )  ->  j  e.  NN )
4241nnrecred 9791 . . . . . . . . . . . . . . . . 17  |-  ( (  T.  /\  j  e. 
Prime )  ->  ( 1  /  j )  e.  RR )
436a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( (  T.  /\  -.  j  e.  Prime )  ->  0  e.  RR )
4442, 43ifclda 3592 . . . . . . . . . . . . . . . 16  |-  (  T. 
->  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
4544trud 1314 . . . . . . . . . . . . . . 15  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR
4645elexi 2797 . . . . . . . . . . . . . 14  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e. 
_V
4739, 9, 46fvmpt 5602 . . . . . . . . . . . . 13  |-  ( j  e.  NN  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
4847adantl 452 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( F `  j )  =  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
4945a1i 10 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
5048, 49eqeltrd 2357 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
5150adantlr 695 . . . . . . . . . 10  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
52 nnrp 10363 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  ->  j  e.  RR+ )
5352adantl 452 . . . . . . . . . . . . . . 15  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  j  e.  RR+ )
5453rpreccld 10400 . . . . . . . . . . . . . 14  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( 1  / 
j )  e.  RR+ )
5554rpge0d 10394 . . . . . . . . . . . . 13  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  (
1  /  j ) )
56 0le0 9827 . . . . . . . . . . . . 13  |-  0  <_  0
57 breq2 4027 . . . . . . . . . . . . . 14  |-  ( ( 1  /  j )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  ->  (
0  <_  ( 1  /  j )  <->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) ) )
58 breq2 4027 . . . . . . . . . . . . . 14  |-  ( 0  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  ->  (
0  <_  0  <->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) ) )
5957, 58ifboth 3596 . . . . . . . . . . . . 13  |-  ( ( 0  <_  ( 1  /  j )  /\  0  <_  0 )  -> 
0  <_  if (
j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
6055, 56, 59sylancl 643 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
6160, 48breqtrrd 4049 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
6261adantlr 695 . . . . . . . . . 10  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j
) )
631, 34, 35, 51, 62climserle 12136 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq  1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq  1
(  +  ,  F
) ) )
6463ralrimiva 2626 . . . . . . . 8  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) )
65 breq2 4027 . . . . . . . . . 10  |-  ( x  =  (  ~~>  `  seq  1 (  +  ,  F ) )  -> 
( (  seq  1
(  +  ,  F
) `  k )  <_  x  <->  (  seq  1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq  1
(  +  ,  F
) ) ) )
6665ralbidv 2563 . . . . . . . . 9  |-  ( x  =  (  ~~>  `  seq  1 (  +  ,  F ) )  -> 
( A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x  <->  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) ) )
6766rspcev 2884 . . . . . . . 8  |-  ( ( (  ~~>  `  seq  1
(  +  ,  F
) )  e.  RR  /\ 
A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq  1 (  +  ,  F ) ) )  ->  E. x  e.  RR  A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x )
6833, 64, 67syl2anc 642 . . . . . . 7  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  E. x  e.  RR  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k
)  <_  x )
69 ffn 5389 . . . . . . . . 9  |-  (  seq  1 (  +  ,  F ) : NN --> RR  ->  seq  1 (  +  ,  F )  Fn  NN )
70 breq1 4026 . . . . . . . . . 10  |-  ( z  =  (  seq  1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq  1 (  +  ,  F ) `  k )  <_  x
) )
7170ralrn 5668 . . . . . . . . 9  |-  (  seq  1 (  +  ,  F )  Fn  NN  ->  ( A. z  e. 
ran  seq  1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq  1
(  +  ,  F
) `  k )  <_  x ) )
7214, 69, 71mp2b 9 . . . . . . . 8  |-  ( A. z  e.  ran  seq  1
(  +  ,  F
) z  <_  x  <->  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k
)  <_  x )
7372rexbii 2568 . . . . . . 7  |-  ( E. x  e.  RR  A. z  e.  ran  seq  1
(  +  ,  F
) z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  (  seq  1 (  +  ,  F ) `  k
)  <_  x )
7468, 73sylibr 203 . . . . . 6  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x )
75 suprcl 9714 . . . . . 6  |-  ( ( ran  seq  1 (  +  ,  F ) 
C_  RR  /\  ran  seq  1 (  +  ,  F )  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x )  ->  sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR )
7617, 26, 74, 75syl3anc 1182 . . . . 5  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  e.  RR )
77 2rp 10359 . . . . . 6  |-  2  e.  RR+
78 rpreccl 10377 . . . . . 6  |-  ( 2  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
7977, 78ax-mp 8 . . . . 5  |-  ( 1  /  2 )  e.  RR+
80 ltsubrp 10385 . . . . 5  |-  ( ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  2
)  e.  RR+ )  ->  ( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  <  sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  ) )
8176, 79, 80sylancl 643 . . . 4  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  -> 
( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  <  sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  ) )
82 rpre 10360 . . . . . . 7  |-  ( ( 1  /  2 )  e.  RR+  ->  ( 1  /  2 )  e.  RR )
8379, 82ax-mp 8 . . . . . 6  |-  ( 1  /  2 )  e.  RR
84 resubcl 9111 . . . . . 6  |-  ( ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  e.  RR )
8576, 83, 84sylancl 643 . . . . 5  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  -> 
( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  e.  RR )
86 suprlub 9716 . . . . 5  |-  ( ( ( ran  seq  1
(  +  ,  F
)  C_  RR  /\  ran  seq  1 (  +  ,  F )  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  seq  1 (  +  ,  F ) z  <_  x )  /\  ( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  e.  RR )  ->  (
( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  <  sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  <->  E. y  e.  ran  seq  1 (  +  ,  F ) ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  y ) )
8717, 26, 74, 85, 86syl31anc 1185 . . . 4  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  -> 
( ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  <  sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  <->  E. y  e.  ran  seq  1 (  +  ,  F ) ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  y ) )
8881, 87mpbid 201 . . 3  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  E. y  e.  ran  seq  1 (  +  ,  F ) ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  y )
89 breq2 4027 . . . . 5  |-  ( y  =  (  seq  1
(  +  ,  F
) `  k )  ->  ( ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
y  <->  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) ) )
9089rexrn 5667 . . . 4  |-  (  seq  1 (  +  ,  F )  Fn  NN  ->  ( E. y  e. 
ran  seq  1 (  +  ,  F ) ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  (
1  /  2 ) )  <  y  <->  E. k  e.  NN  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) ) )
9114, 69, 90mp2b 9 . . 3  |-  ( E. y  e.  ran  seq  1 (  +  ,  F ) ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  y  <->  E. k  e.  NN  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) )
9288, 91sylib 188 . 2  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  E. k  e.  NN  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  (
1  /  2 ) )  <  (  seq  1 (  +  ,  F ) `  k
) )
93 2re 9815 . . . . . 6  |-  2  e.  RR
94 2nn 9877 . . . . . . . . 9  |-  2  e.  NN
95 nnmulcl 9769 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
9694, 34, 95sylancr 644 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  e.  NN )
9796peano2nnd 9763 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  e.  NN )
9897nnnn0d 10018 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  e.  NN0 )
99 reexpcl 11120 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( 2  x.  k
)  +  1 ) )  e.  RR )
10093, 98, 99sylancr 644 . . . . 5  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  RR )
101100ltnrd 8953 . . . 4  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  -.  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  < 
( 2 ^ (
( 2  x.  k
)  +  1 ) ) )
10234adantr 451 . . . . . . 7  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  k  e.  NN )
103 peano2nn 9758 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
104103adantl 452 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  NN )
105104nnnn0d 10018 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  NN0 )
106 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( k  +  1 )  e.  NN0 )  ->  ( 2 ^ (
k  +  1 ) )  e.  NN )
10794, 105, 106sylancr 644 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  1 ) )  e.  NN )
108107nnsqcld 11265 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) ) ^
2 )  e.  NN )
109108adantr 451 . . . . . . 7  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  (
( 2 ^ (
k  +  1 ) ) ^ 2 )  e.  NN )
110 breq1 4026 . . . . . . . . . . 11  |-  ( p  =  w  ->  (
p  ||  r  <->  w  ||  r
) )
111110notbid 285 . . . . . . . . . 10  |-  ( p  =  w  ->  ( -.  p  ||  r  <->  -.  w  ||  r ) )
112111cbvralv 2764 . . . . . . . . 9  |-  ( A. p  e.  ( Prime  \  ( 1 ... k
) )  -.  p  ||  r  <->  A. w  e.  ( Prime  \  ( 1 ... k ) )  -.  w  ||  r
)
113 breq2 4027 . . . . . . . . . . 11  |-  ( r  =  n  ->  (
w  ||  r  <->  w  ||  n
) )
114113notbid 285 . . . . . . . . . 10  |-  ( r  =  n  ->  ( -.  w  ||  r  <->  -.  w  ||  n ) )
115114ralbidv 2563 . . . . . . . . 9  |-  ( r  =  n  ->  ( A. w  e.  ( Prime  \  ( 1 ... k ) )  -.  w  ||  r  <->  A. w  e.  ( Prime  \  (
1 ... k ) )  -.  w  ||  n
) )
116112, 115syl5bb 248 . . . . . . . 8  |-  ( r  =  n  ->  ( A. p  e.  ( Prime  \  ( 1 ... k ) )  -.  p  ||  r  <->  A. w  e.  ( Prime  \  (
1 ... k ) )  -.  w  ||  n
) )
117116cbvrabv 2787 . . . . . . 7  |-  { r  e.  ( 1 ... ( ( 2 ^ ( k  +  1 ) ) ^ 2 ) )  |  A. p  e.  ( Prime  \  ( 1 ... k
) )  -.  p  ||  r }  =  {
n  e.  ( 1 ... ( ( 2 ^ ( k  +  1 ) ) ^
2 ) )  | 
A. w  e.  ( Prime  \  ( 1 ... k ) )  -.  w  ||  n }
118 simpll 730 . . . . . . 7  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )
119 eleq1 2343 . . . . . . . . . 10  |-  ( m  =  j  ->  (
m  e.  Prime  <->  j  e.  Prime ) )
120 oveq2 5866 . . . . . . . . . 10  |-  ( m  =  j  ->  (
1  /  m )  =  ( 1  / 
j ) )
121 eqidd 2284 . . . . . . . . . 10  |-  ( m  =  j  ->  0  =  0 )
122119, 120, 121ifbieq12d 3587 . . . . . . . . 9  |-  ( m  =  j  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
123122cbvsumv 12169 . . . . . . . 8  |-  sum_ m  e.  ( ZZ>= `  ( k  +  1 ) ) if ( m  e. 
Prime ,  ( 1  /  m ) ,  0 )  =  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )
124 simpr 447 . . . . . . . 8  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  <  (
1  /  2 ) )
125123, 124syl5eqbr 4056 . . . . . . 7  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  sum_ m  e.  ( ZZ>= `  ( k  +  1 ) ) if ( m  e. 
Prime ,  ( 1  /  m ) ,  0 )  <  (
1  /  2 ) )
126 eqid 2283 . . . . . . 7  |-  ( w  e.  NN  |->  { n  e.  ( 1 ... (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  |  ( w  e.  Prime  /\  w  ||  n ) } )  =  ( w  e.  NN  |->  { n  e.  ( 1 ... (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  |  ( w  e.  Prime  /\  w  ||  n ) } )
1279, 102, 109, 117, 118, 125, 126prmreclem5 12967 . . . . . 6  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  < 
( 1  /  2
) )  ->  (
( ( 2 ^ ( k  +  1 ) ) ^ 2 )  /  2 )  <  ( ( 2 ^ k )  x.  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) ) ) )
128127ex 423 . . . . 5  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  <  (
1  /  2 )  ->  ( ( ( 2 ^ ( k  +  1 ) ) ^ 2 )  / 
2 )  <  (
( 2 ^ k
)  x.  ( sqr `  ( ( 2 ^ ( k  +  1 ) ) ^ 2 ) ) ) ) )
129 eqid 2283 . . . . . . . . 9  |-  ( ZZ>= `  ( k  +  1 ) )  =  (
ZZ>= `  ( k  +  1 ) )
130104nnzd 10116 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  ZZ )
1311uztrn2 10245 . . . . . . . . . . 11  |-  ( ( ( k  +  1 )  e.  NN  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  -> 
j  e.  NN )
132104, 131sylan 457 . . . . . . . . . 10  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  j  e.  NN )
133132, 47syl 15 . . . . . . . . 9  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  ( F `  j )  =  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
13445a1i 10 . . . . . . . . 9  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  if (
j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
135 simpl 443 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq  1 (  +  ,  F )  e.  dom  ~~>  )
13647adantl 452 . . . . . . . . . . . 12  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
13745recni 8849 . . . . . . . . . . . . 13  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  CC
138137a1i 10 . . . . . . . . . . . 12  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  CC )
139136, 138eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  CC )
1401, 104, 139iserex 12130 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq  1
(  +  ,  F
)  e.  dom  ~~>  <->  seq  ( k  +  1 ) (  +  ,  F )  e.  dom  ~~>  ) )
141135, 140mpbid 201 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq  ( k  +  1 ) (  +  ,  F )  e.  dom  ~~>  )
142129, 130, 133, 134, 141isumrecl 12228 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  (
ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
14383a1i 10 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 1  / 
2 )  e.  RR )
144 elfznn 10819 . . . . . . . . . . . 12  |-  ( j  e.  ( 1 ... k )  ->  j  e.  NN )
145144adantl 452 . . . . . . . . . . 11  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  j  e.  NN )
146145, 47syl 15 . . . . . . . . . 10  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
14734, 1syl6eleq 2373 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  (
ZZ>= `  1 ) )
148137a1i 10 . . . . . . . . . 10  |-  ( ( (  seq  1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  CC )
149146, 147, 148fsumser 12203 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  ( 1 ... k ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  =  (  seq  1 (  +  ,  F ) `  k ) )
150149, 32eqeltrd 2357 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  ( 1 ... k ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
151142, 143, 150ltadd2d 8972 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  <  (
1  /  2 )  <-> 
( sum_ j  e.  ( 1 ... k ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  +  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )  <  ( sum_ j  e.  ( 1 ... k
) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  ( 1  /  2 ) ) ) )
1521, 129, 104, 136, 138, 135isumsplit 12299 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  =  ( sum_ j  e.  ( 1 ... (
( k  +  1 )  -  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) ) )
153 nncn 9754 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  CC )
154153adantl 452 . . . . . . . . . . . . 13  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  CC )
155 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
156 pncan 9057 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  +  1 )  -  1 )  =  k )
157154, 155, 156sylancl 643 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  - 
1 )  =  k )
158157oveq2d 5874 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 1 ... ( ( k  +  1 )  -  1 ) )  =  ( 1 ... k ) )
159158sumeq1d 12174 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  ( 1 ... ( ( k  +  1 )  -  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  =  sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
160159oveq1d 5873 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  ( 1 ... (
( k  +  1 )  -  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )  =  ( sum_ j  e.  ( 1 ... k
) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) ) )
161152, 160eqtrd 2315 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  =  ( sum_ j  e.  ( 1 ... k
) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) ) )
162161breq1d 4033 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  <  ( sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  +  ( 1  / 
2 ) )  <->  ( sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  +  sum_ j  e.  (
ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )  <  ( sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  +  ( 1  / 
2 ) ) ) )
163151, 162bitr4d 247 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  <  (
1  /  2 )  <->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  <  ( sum_ j  e.  ( 1 ... k
) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  ( 1  /  2 ) ) ) )
164 eqid 2283 . . . . . . . . . 10  |-  seq  1
(  +  ,  F
)  =  seq  1
(  +  ,  F
)
1651, 164, 27, 48, 49, 60, 68isumsup 12306 . . . . . . . . 9  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  =  sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )
)
166165, 76eqeltrd 2357 . . . . . . . 8  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
167166adantr 451 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
168167, 143, 150ltsubaddd 9368 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  -  ( 1  / 
2 ) )  <  sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  <->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  <  ( sum_ j  e.  ( 1 ... k
) if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  +  ( 1  /  2 ) ) ) )
169165adantr 451 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  =  sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )
)
170169oveq1d 5873 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  -  (
1  /  2 ) )  =  ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) ) )
171170, 149breq12d 4036 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  -  ( 1  / 
2 ) )  <  sum_ j  e.  ( 1 ... k ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  <-> 
( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) ) )
172163, 168, 1713bitr2d 272 . . . . 5  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sum_ j  e.  ( ZZ>= `  ( k  +  1 ) ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  <  (
1  /  2 )  <-> 
( sup ( ran 
seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) ) )
173 2cn 9816 . . . . . . . . . . . . 13  |-  2  e.  CC
174173a1i 10 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  2  e.  CC )
175155a1i 10 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  1  e.  CC )
176174, 154, 175adddid 8859 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  ( k  +  1 ) )  =  ( ( 2  x.  k
)  +  ( 2  x.  1 ) ) )
177104nncnd 9762 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  CC )
178 mulcom 8823 . . . . . . . . . . . 12  |-  ( ( ( k  +  1 )  e.  CC  /\  2  e.  CC )  ->  ( ( k  +  1 )  x.  2 )  =  ( 2  x.  ( k  +  1 ) ) )
179177, 173, 178sylancl 643 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  2 )  =  ( 2  x.  ( k  +  1 ) ) )
18096nncnd 9762 . . . . . . . . . . . . 13  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  e.  CC )
181180, 175, 175addassd 8857 . . . . . . . . . . . 12  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2  x.  k )  +  1 )  +  1 )  =  ( ( 2  x.  k
)  +  ( 1  +  1 ) ) )
1821552timesi 9845 . . . . . . . . . . . . 13  |-  ( 2  x.  1 )  =  ( 1  +  1 )
183182oveq2i 5869 . . . . . . . . . . . 12  |-  ( ( 2  x.  k )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  k )  +  ( 1  +  1 ) )
184181, 183syl6eqr 2333 . . . . . . . . . . 11  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2  x.  k )  +  1 )  +  1 )  =  ( ( 2  x.  k
)  +  ( 2  x.  1 ) ) )
185176, 179, 1843eqtr4d 2325 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  2 )  =  ( ( ( 2  x.  k )  +  1 )  +  1 ) )
186185oveq2d 5874 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( k  +  1 )  x.  2 ) )  =  ( 2 ^ ( ( ( 2  x.  k
)  +  1 )  +  1 ) ) )
187 2nn0 9982 . . . . . . . . . . 11  |-  2  e.  NN0
188187a1i 10 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  2  e.  NN0 )
189174, 188, 105expmuld 11248 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( k  +  1 )  x.  2 ) )  =  ( ( 2 ^ (
k  +  1 ) ) ^ 2 ) )
190 expp1 11110 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( ( 2  x.  k )  +  1 )  +  1 ) )  =  ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 ) )
191173, 98, 190sylancr 644 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( ( 2  x.  k )  +  1 )  +  1 ) )  =  ( ( 2 ^ (
( 2  x.  k
)  +  1 ) )  x.  2 ) )
192186, 189, 1913eqtr3d 2323 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) ) ^
2 )  =  ( ( 2 ^ (
( 2  x.  k
)  +  1 ) )  x.  2 ) )
193192oveq1d 5873 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( k  +  1 ) ) ^ 2 )  / 
2 )  =  ( ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  /  2 ) )
194 expcl 11121 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( 2  x.  k
)  +  1 ) )  e.  CC )
195173, 98, 194sylancr 644 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  CC )
196 2ne0 9829 . . . . . . . . 9  |-  2  =/=  0
197 divcan4 9449 . . . . . . . . 9  |-  ( ( ( 2 ^ (
( 2  x.  k
)  +  1 ) )  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  /  2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
198173, 196, 197mp3an23 1269 . . . . . . . 8  |-  ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  CC  ->  (
( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  /  2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
199195, 198syl 15 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  / 
2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
200193, 199eqtrd 2315 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( k  +  1 ) ) ^ 2 )  / 
2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
201 nnnn0 9972 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
202201adantl 452 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  NN0 )
203174, 105, 202expaddd 11247 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  ( k  +  1 ) ) )  =  ( ( 2 ^ k
)  x.  ( 2 ^ ( k  +  1 ) ) ) )
2041542timesd 9954 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  =  ( k  +  k ) )
205204oveq1d 5873 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  =  ( ( k  +  k )  +  1 ) )
206154, 154, 175addassd 8857 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  k )  +  1 )  =  ( k  +  ( k  +  1 ) ) )
207205, 206eqtrd 2315 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  =  ( k  +  ( k  +  1 ) ) )
208207oveq2d 5874 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  =  ( 2 ^ ( k  +  ( k  +  1 ) ) ) )
209107nnrpd 10389 . . . . . . . . . 10  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  1 ) )  e.  RR+ )
210209rprege0d 10397 . . . . . . . . 9  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) )  e.  RR  /\  0  <_ 
( 2 ^ (
k  +  1 ) ) ) )
211 sqrsq 11755 . . . . . . . . 9  |-  ( ( ( 2 ^ (
k  +  1 ) )  e.  RR  /\  0  <_  ( 2 ^ ( k  +  1 ) ) )  -> 
( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  =  ( 2 ^ ( k  +  1 ) ) )
212210, 211syl 15 . . . . . . . 8  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  =  ( 2 ^ ( k  +  1 ) ) )
213212oveq2d 5874 . . . . . . 7  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ k )  x.  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) ) )  =  ( ( 2 ^ k
)  x.  ( 2 ^ ( k  +  1 ) ) ) )
214203, 208, 2133eqtr4rd 2326 . . . . . 6  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ k )  x.  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) ) )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
215200, 214breq12d 4036 . . . . 5  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( ( 2 ^ (
k  +  1 ) ) ^ 2 )  /  2 )  < 
( ( 2 ^ k )  x.  ( sqr `  ( ( 2 ^ ( k  +  1 ) ) ^
2 ) ) )  <-> 
( 2 ^ (
( 2  x.  k
)  +  1 ) )  <  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) ) )
216128, 172, 2153imtr3d 258 . . . 4  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( sup ( ran  seq  1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  (  seq  1
(  +  ,  F
) `  k )  ->  ( 2 ^ (
( 2  x.  k
)  +  1 ) )  <  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) ) )
217101, 216mtod 168 . . 3  |-  ( (  seq  1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  -.  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq  1 (  +  ,  F ) `
 k ) )
218217nrexdv 2646 . 2  |-  (  seq  1 (  +  ,  F )  e.  dom  ~~>  ->  -.  E. k  e.  NN  ( sup ( ran  seq  1 (  +  ,  F ) ,  RR ,  <  )  -  (
1  /  2 ) )  <  (  seq  1 (  +  ,  F ) `  k
) )
21992, 218pm2.65i 165 1  |-  -.  seq  1 (  +  ,  F )  e.  dom  ~~>
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   (/)c0 3455   ifcif 3565   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    seq cseq 11046   ^cexp 11104   sqrcsqr 11718    ~~> cli 11958   sum_csu 12158    || cdivides 12531   Primecprime 12758
This theorem is referenced by:  prmrec  12969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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