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Theorem prmreclem6 13974
Description: Lemma for prmrec 13975. 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 13973 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 10888 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10669 . . . . . . . . . 10  |-  ( T. 
->  1  e.  ZZ )
3 nnrecre 10350 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
1  /  n )  e.  RR )
43adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  n  e.  NN )  ->  (
1  /  n )  e.  RR )
5 0re 9378 . . . . . . . . . . . . 13  |-  0  e.  RR
6 ifcl 3824 . . . . . . . . . . . . 13  |-  ( ( ( 1  /  n
)  e.  RR  /\  0  e.  RR )  ->  if ( n  e. 
Prime ,  ( 1  /  n ) ,  0 )  e.  RR )
74, 5, 6sylancl 662 . . . . . . . . . . . 12  |-  ( ( T.  /\  n  e.  NN )  ->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 )  e.  RR )
8 prmrec.1 . . . . . . . . . . . 12  |-  F  =  ( n  e.  NN  |->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 ) )
97, 8fmptd 5860 . . . . . . . . . . 11  |-  ( T. 
->  F : NN --> RR )
109ffvelrnda 5836 . . . . . . . . . 10  |-  ( ( T.  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
111, 2, 10serfre 11827 . . . . . . . . 9  |-  ( T. 
->  seq 1 (  +  ,  F ) : NN --> RR )
1211trud 1378 . . . . . . . 8  |-  seq 1
(  +  ,  F
) : NN --> RR
13 frn 5558 . . . . . . . 8  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  ran  seq 1
(  +  ,  F
)  C_  RR )
1412, 13ax-mp 5 . . . . . . 7  |-  ran  seq 1 (  +  ,  F )  C_  RR
1514a1i 11 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  ran  seq 1 (  +  ,  F )  C_  RR )
16 1nn 10325 . . . . . . . . 9  |-  1  e.  NN
1712fdmi 5557 . . . . . . . . 9  |-  dom  seq 1 (  +  ,  F )  =  NN
1816, 17eleqtrri 2510 . . . . . . . 8  |-  1  e.  dom  seq 1 (  +  ,  F )
19 ne0i 3636 . . . . . . . . 9  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  dom  seq 1
(  +  ,  F
)  =/=  (/) )
20 dm0rn0 5048 . . . . . . . . . 10  |-  ( dom 
seq 1 (  +  ,  F )  =  (/) 
<->  ran  seq 1 (  +  ,  F )  =  (/) )
2120necon3bii 2634 . . . . . . . . 9  |-  ( dom 
seq 1 (  +  ,  F )  =/=  (/) 
<->  ran  seq 1 (  +  ,  F )  =/=  (/) )
2219, 21sylib 196 . . . . . . . 8  |-  ( 1  e.  dom  seq 1
(  +  ,  F
)  ->  ran  seq 1
(  +  ,  F
)  =/=  (/) )
2318, 22ax-mp 5 . . . . . . 7  |-  ran  seq 1 (  +  ,  F )  =/=  (/)
2423a1i 11 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  ran  seq 1 (  +  ,  F )  =/=  (/) )
25 1zzd 10669 . . . . . . . . 9  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  -> 
1  e.  ZZ )
26 climdm 13024 . . . . . . . . . 10  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  <->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2726biimpi 194 . . . . . . . . 9  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
2812a1i 11 . . . . . . . . . 10  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  seq 1 (  +  ,  F ) : NN --> RR )
2928ffvelrnda 5836 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq 1
(  +  ,  F
) `  k )  e.  RR )
301, 25, 27, 29climrecl 13053 . . . . . . . 8  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  -> 
(  ~~>  `  seq 1
(  +  ,  F
) )  e.  RR )
31 simpr 461 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  NN )
3227adantr 465 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq 1 (  +  ,  F )  ~~>  (  ~~>  `  seq 1 (  +  ,  F ) ) )
33 eleq1 2497 . . . . . . . . . . . . . . 15  |-  ( n  =  j  ->  (
n  e.  Prime  <->  j  e.  Prime ) )
34 oveq2 6094 . . . . . . . . . . . . . . 15  |-  ( n  =  j  ->  (
1  /  n )  =  ( 1  / 
j ) )
3533, 34ifbieq1d 3805 . . . . . . . . . . . . . 14  |-  ( n  =  j  ->  if ( n  e.  Prime ,  ( 1  /  n
) ,  0 )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
36 prmnn 13758 . . . . . . . . . . . . . . . . . . 19  |-  ( j  e.  Prime  ->  j  e.  NN )
3736adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( T.  /\  j  e. 
Prime )  ->  j  e.  NN )
3837nnrecred 10359 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  j  e. 
Prime )  ->  ( 1  /  j )  e.  RR )
395a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  -.  j  e.  Prime )  ->  0  e.  RR )
4038, 39ifclda 3814 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
4140trud 1378 . . . . . . . . . . . . . . 15  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR
4241elexi 2976 . . . . . . . . . . . . . 14  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e. 
_V
4335, 8, 42fvmpt 5767 . . . . . . . . . . . . 13  |-  ( j  e.  NN  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
4443adantl 466 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( F `  j )  =  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
4541a1i 11 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
4644, 45eqeltrd 2511 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
4746adantlr 714 . . . . . . . . . 10  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  RR )
48 nnrp 10992 . . . . . . . . . . . . . . . 16  |-  ( j  e.  NN  ->  j  e.  RR+ )
4948adantl 466 . . . . . . . . . . . . . . 15  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  j  e.  RR+ )
5049rpreccld 11029 . . . . . . . . . . . . . 14  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  ( 1  / 
j )  e.  RR+ )
5150rpge0d 11023 . . . . . . . . . . . . 13  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  (
1  /  j ) )
52 0le0 10403 . . . . . . . . . . . . 13  |-  0  <_  0
53 breq2 4289 . . . . . . . . . . . . . 14  |-  ( ( 1  /  j )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  ->  (
0  <_  ( 1  /  j )  <->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) ) )
54 breq2 4289 . . . . . . . . . . . . . 14  |-  ( 0  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  ->  (
0  <_  0  <->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) ) )
5553, 54ifboth 3818 . . . . . . . . . . . . 13  |-  ( ( 0  <_  ( 1  /  j )  /\  0  <_  0 )  -> 
0  <_  if (
j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
5651, 52, 55sylancl 662 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
5756, 44breqtrrd 4311 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  j  e.  NN )  ->  0  <_  ( F `  j )
)
5857adantlr 714 . . . . . . . . . 10  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  0  <_  ( F `  j
) )
591, 31, 32, 47, 58climserle 13132 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) )
6059ralrimiva 2793 . . . . . . . 8  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k )  <_  (  ~~>  ` 
seq 1 (  +  ,  F ) ) )
61 breq2 4289 . . . . . . . . . 10  |-  ( x  =  (  ~~>  `  seq 1 (  +  ,  F ) )  -> 
( (  seq 1
(  +  ,  F
) `  k )  <_  x  <->  (  seq 1
(  +  ,  F
) `  k )  <_  (  ~~>  `  seq 1
(  +  ,  F
) ) ) )
6261ralbidv 2729 . . . . . . . . 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 ) ) ) )
6362rspcev 3066 . . . . . . . 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 )
6430, 60, 63syl2anc 661 . . . . . . 7  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  E. x  e.  RR  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k
)  <_  x )
65 ffn 5552 . . . . . . . . 9  |-  (  seq 1 (  +  ,  F ) : NN --> RR  ->  seq 1 (  +  ,  F )  Fn  NN )
66 breq1 4288 . . . . . . . . . 10  |-  ( z  =  (  seq 1
(  +  ,  F
) `  k )  ->  ( z  <_  x  <->  (  seq 1 (  +  ,  F ) `  k )  <_  x
) )
6766ralrn 5839 . . . . . . . . 9  |-  (  seq 1 (  +  ,  F )  Fn  NN  ->  ( A. z  e. 
ran  seq 1 (  +  ,  F ) z  <_  x  <->  A. k  e.  NN  (  seq 1
(  +  ,  F
) `  k )  <_  x ) )
6812, 65, 67mp2b 10 . . . . . . . 8  |-  ( A. z  e.  ran  seq 1
(  +  ,  F
) z  <_  x  <->  A. k  e.  NN  (  seq 1 (  +  ,  F ) `  k
)  <_  x )
6968rexbii 2734 . . . . . . 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 )
7064, 69sylibr 212 . . . . . 6  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  E. x  e.  RR  A. z  e.  ran  seq 1 (  +  ,  F ) z  <_  x )
71 suprcl 10282 . . . . . 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 )
7215, 24, 70, 71syl3anc 1218 . . . . 5  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  sup ( ran  seq 1
(  +  ,  F
) ,  RR ,  <  )  e.  RR )
73 2rp 10988 . . . . . 6  |-  2  e.  RR+
74 rpreccl 11006 . . . . . 6  |-  ( 2  e.  RR+  ->  ( 1  /  2 )  e.  RR+ )
7573, 74ax-mp 5 . . . . 5  |-  ( 1  /  2 )  e.  RR+
76 ltsubrp 11014 . . . . 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 ,  <  ) )
7772, 75, 76sylancl 662 . . . 4  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  -> 
( sup ( ran 
seq 1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  <  sup ( ran  seq 1
(  +  ,  F
) ,  RR ,  <  ) )
78 halfre 10532 . . . . . 6  |-  ( 1  /  2 )  e.  RR
79 resubcl 9665 . . . . . 6  |-  ( ( sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  2
)  e.  RR )  ->  ( sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  e.  RR )
8072, 78, 79sylancl 662 . . . . 5  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  -> 
( sup ( ran 
seq 1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  e.  RR )
81 suprlub 10284 . . . . 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 ) )
8215, 24, 70, 80, 81syl31anc 1221 . . . 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 ) )
8377, 82mpbid 210 . . 3  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  E. y  e.  ran  seq 1 (  +  ,  F ) ( sup ( ran  seq 1
(  +  ,  F
) ,  RR ,  <  )  -  ( 1  /  2 ) )  <  y )
84 breq2 4289 . . . . 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 ) ) )
8584rexrn 5838 . . . 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 ) ) )
8612, 65, 85mp2b 10 . . 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 ) )
8783, 86sylib 196 . 2  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  E. k  e.  NN  ( sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  -  (
1  /  2 ) )  <  (  seq 1 (  +  ,  F ) `  k
) )
88 2re 10383 . . . . . 6  |-  2  e.  RR
89 2nn 10471 . . . . . . . . 9  |-  2  e.  NN
90 nnmulcl 10337 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  k  e.  NN )  ->  ( 2  x.  k
)  e.  NN )
9189, 31, 90sylancr 663 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  e.  NN )
9291peano2nnd 10331 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  e.  NN )
9392nnnn0d 10628 . . . . . 6  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  e.  NN0 )
94 reexpcl 11874 . . . . . 6  |-  ( ( 2  e.  RR  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( 2  x.  k
)  +  1 ) )  e.  RR )
9588, 93, 94sylancr 663 . . . . 5  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  RR )
9695ltnrd 9500 . . . 4  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  -.  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  < 
( 2 ^ (
( 2  x.  k
)  +  1 ) ) )
9731adantr 465 . . . . . . 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 )
98 peano2nn 10326 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
k  +  1 )  e.  NN )
9998adantl 466 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  NN )
10099nnnn0d 10628 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  NN0 )
101 nnexpcl 11870 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( k  +  1 )  e.  NN0 )  ->  ( 2 ^ (
k  +  1 ) )  e.  NN )
10289, 100, 101sylancr 663 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  1 ) )  e.  NN )
103102nnsqcld 12020 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) ) ^
2 )  e.  NN )
104103adantr 465 . . . . . . 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 )
105 breq1 4288 . . . . . . . . . . 11  |-  ( p  =  w  ->  (
p  ||  r  <->  w  ||  r
) )
106105notbid 294 . . . . . . . . . 10  |-  ( p  =  w  ->  ( -.  p  ||  r  <->  -.  w  ||  r ) )
107106cbvralv 2941 . . . . . . . . 9  |-  ( A. p  e.  ( Prime  \  ( 1 ... k
) )  -.  p  ||  r  <->  A. w  e.  ( Prime  \  ( 1 ... k ) )  -.  w  ||  r
)
108 breq2 4289 . . . . . . . . . . 11  |-  ( r  =  n  ->  (
w  ||  r  <->  w  ||  n
) )
109108notbid 294 . . . . . . . . . 10  |-  ( r  =  n  ->  ( -.  w  ||  r  <->  -.  w  ||  n ) )
110109ralbidv 2729 . . . . . . . . 9  |-  ( r  =  n  ->  ( A. w  e.  ( Prime  \  ( 1 ... k ) )  -.  w  ||  r  <->  A. w  e.  ( Prime  \  (
1 ... k ) )  -.  w  ||  n
) )
111107, 110syl5bb 257 . . . . . . . 8  |-  ( r  =  n  ->  ( A. p  e.  ( Prime  \  ( 1 ... k ) )  -.  p  ||  r  <->  A. w  e.  ( Prime  \  (
1 ... k ) )  -.  w  ||  n
) )
112111cbvrabv 2965 . . . . . . 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 }
113 simpll 753 . . . . . . 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  ~~>  )
114 eleq1 2497 . . . . . . . . . 10  |-  ( m  =  j  ->  (
m  e.  Prime  <->  j  e.  Prime ) )
115 oveq2 6094 . . . . . . . . . 10  |-  ( m  =  j  ->  (
1  /  m )  =  ( 1  / 
j ) )
116114, 115ifbieq1d 3805 . . . . . . . . 9  |-  ( m  =  j  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 ) )
117116cbvsumv 13165 . . . . . . . 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 )
118 simpr 461 . . . . . . . 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 ) )
119117, 118syl5eqbr 4318 . . . . . . 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 ) )
120 eqid 2437 . . . . . . 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 ) } )
1218, 97, 104, 112, 113, 119, 120prmreclem5 13973 . . . . . 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 ) ) ) )
122121ex 434 . . . . 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 ) ) ) ) )
123 eqid 2437 . . . . . . . . 9  |-  ( ZZ>= `  ( k  +  1 ) )  =  (
ZZ>= `  ( k  +  1 ) )
12499nnzd 10738 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  ZZ )
125 eluznn 10917 . . . . . . . . . . 11  |-  ( ( ( k  +  1 )  e.  NN  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  -> 
j  e.  NN )
12699, 125sylan 471 . . . . . . . . . 10  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  j  e.  NN )
127126, 43syl 16 . . . . . . . . 9  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  ( F `  j )  =  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 ) )
12841a1i 11 . . . . . . . . 9  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( ZZ>= `  ( k  +  1 ) ) )  ->  if (
j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
129 simpl 457 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq 1 (  +  ,  F )  e. 
dom 
~~>  )
13043adantl 466 . . . . . . . . . . . 12  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
13141recni 9390 . . . . . . . . . . . . 13  |-  if ( j  e.  Prime ,  ( 1  /  j ) ,  0 )  e.  CC
132131a1i 11 . . . . . . . . . . . 12  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  CC )
133130, 132eqeltrd 2511 . . . . . . . . . . 11  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  NN )  ->  ( F `  j )  e.  CC )
1341, 99, 133iserex 13126 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  (  seq 1
(  +  ,  F
)  e.  dom  ~~>  <->  seq (
k  +  1 ) (  +  ,  F
)  e.  dom  ~~>  ) )
135129, 134mpbid 210 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  seq ( k  +  1 ) (  +  ,  F )  e. 
dom 
~~>  )
136123, 124, 127, 128, 135isumrecl 13224 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  (
ZZ>= `  ( k  +  1 ) ) if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
13778a1i 11 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 1  / 
2 )  e.  RR )
138 elfznn 11470 . . . . . . . . . . . 12  |-  ( j  e.  ( 1 ... k )  ->  j  e.  NN )
139138adantl 466 . . . . . . . . . . 11  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  j  e.  NN )
140139, 43syl 16 . . . . . . . . . 10  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  ( F `  j )  =  if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 ) )
14131, 1syl6eleq 2527 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  (
ZZ>= `  1 ) )
142131a1i 11 . . . . . . . . . 10  |-  ( ( (  seq 1 (  +  ,  F )  e.  dom  ~~>  /\  k  e.  NN )  /\  j  e.  ( 1 ... k
) )  ->  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  CC )
143140, 141, 142fsumser 13199 . . . . . . . . 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 ) )
144143, 29eqeltrd 2511 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  ( 1 ... k ) if ( j  e. 
Prime ,  ( 1  /  j ) ,  0 )  e.  RR )
145136, 137, 144ltadd2d 9519 . . . . . . 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 ) ) ) )
1461, 123, 99, 130, 132, 129isumsplit 13295 . . . . . . . . 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 ) ) )
147 nncn 10322 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  CC )
148147adantl 466 . . . . . . . . . . . . 13  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  CC )
149 ax-1cn 9332 . . . . . . . . . . . . 13  |-  1  e.  CC
150 pncan 9608 . . . . . . . . . . . . 13  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  +  1 )  -  1 )  =  k )
151148, 149, 150sylancl 662 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  - 
1 )  =  k )
152151oveq2d 6102 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 1 ... ( ( k  +  1 )  -  1 ) )  =  ( 1 ... k ) )
153152sumeq1d 13170 . . . . . . . . . 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 ) )
154153oveq1d 6101 . . . . . . . . 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 ) ) )
155146, 154eqtrd 2469 . . . . . . . 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 ) ) )
156155breq1d 4295 . . . . . . 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 ) ) ) )
157145, 156bitr4d 256 . . . . . 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 ) ) ) )
158 eqid 2437 . . . . . . . . . 10  |-  seq 1
(  +  ,  F
)  =  seq 1
(  +  ,  F
)
1591, 158, 25, 44, 45, 56, 64isumsup 13302 . . . . . . . . 9  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  =  sup ( ran 
seq 1 (  +  ,  F ) ,  RR ,  <  )
)
160159, 72eqeltrd 2511 . . . . . . . 8  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
161160adantr 465 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  sum_ j  e.  NN  if ( j  e.  Prime ,  ( 1  /  j
) ,  0 )  e.  RR )
162161, 137, 144ltsubaddd 9927 . . . . . 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 ) ) ) )
163159adantr 465 . . . . . . . 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 ,  <  )
)
164163oveq1d 6101 . . . . . . 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 ) ) )
165164, 143breq12d 4298 . . . . . 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 ) ) )
166157, 162, 1653bitr2d 281 . . . . 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 ) ) )
167 2cn 10384 . . . . . . . . . . . . 13  |-  2  e.  CC
168167a1i 11 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  2  e.  CC )
169149a1i 11 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  1  e.  CC )
170168, 148, 169adddid 9402 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  ( k  +  1 ) )  =  ( ( 2  x.  k
)  +  ( 2  x.  1 ) ) )
17199nncnd 10330 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( k  +  1 )  e.  CC )
172 mulcom 9360 . . . . . . . . . . . 12  |-  ( ( ( k  +  1 )  e.  CC  /\  2  e.  CC )  ->  ( ( k  +  1 )  x.  2 )  =  ( 2  x.  ( k  +  1 ) ) )
173171, 167, 172sylancl 662 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  2 )  =  ( 2  x.  ( k  +  1 ) ) )
17491nncnd 10330 . . . . . . . . . . . . 13  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  e.  CC )
175174, 169, 169addassd 9400 . . . . . . . . . . . 12  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2  x.  k )  +  1 )  +  1 )  =  ( ( 2  x.  k
)  +  ( 1  +  1 ) ) )
1761492timesi 10434 . . . . . . . . . . . . 13  |-  ( 2  x.  1 )  =  ( 1  +  1 )
177176oveq2i 6097 . . . . . . . . . . . 12  |-  ( ( 2  x.  k )  +  ( 2  x.  1 ) )  =  ( ( 2  x.  k )  +  ( 1  +  1 ) )
178175, 177syl6eqr 2487 . . . . . . . . . . 11  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2  x.  k )  +  1 )  +  1 )  =  ( ( 2  x.  k
)  +  ( 2  x.  1 ) ) )
179170, 173, 1783eqtr4d 2479 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  1 )  x.  2 )  =  ( ( ( 2  x.  k )  +  1 )  +  1 ) )
180179oveq2d 6102 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( k  +  1 )  x.  2 ) )  =  ( 2 ^ ( ( ( 2  x.  k
)  +  1 )  +  1 ) ) )
181 2nn0 10588 . . . . . . . . . . 11  |-  2  e.  NN0
182181a1i 11 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  2  e.  NN0 )
183168, 182, 100expmuld 12003 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( k  +  1 )  x.  2 ) )  =  ( ( 2 ^ (
k  +  1 ) ) ^ 2 ) )
184 expp1 11864 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( ( 2  x.  k )  +  1 )  +  1 ) )  =  ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 ) )
185167, 93, 184sylancr 663 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( ( 2  x.  k )  +  1 )  +  1 ) )  =  ( ( 2 ^ (
( 2  x.  k
)  +  1 ) )  x.  2 ) )
186180, 183, 1853eqtr3d 2477 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) ) ^
2 )  =  ( ( 2 ^ (
( 2  x.  k
)  +  1 ) )  x.  2 ) )
187186oveq1d 6101 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( k  +  1 ) ) ^ 2 )  / 
2 )  =  ( ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  /  2 ) )
188 expcl 11875 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( 2  x.  k )  +  1 )  e.  NN0 )  ->  ( 2 ^ (
( 2  x.  k
)  +  1 ) )  e.  CC )
189167, 93, 188sylancr 663 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  CC )
190 2ne0 10406 . . . . . . . . 9  |-  2  =/=  0
191 divcan4 10011 . . . . . . . . 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 ) ) )
192167, 190, 191mp3an23 1306 . . . . . . . 8  |-  ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  e.  CC  ->  (
( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  /  2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
193189, 192syl 16 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( ( 2  x.  k )  +  1 ) )  x.  2 )  / 
2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
194187, 193eqtrd 2469 . . . . . 6  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( ( 2 ^ ( k  +  1 ) ) ^ 2 )  / 
2 )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
195 nnnn0 10578 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
196195adantl 466 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  k  e.  NN0 )
197168, 100, 196expaddd 12002 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  ( k  +  1 ) ) )  =  ( ( 2 ^ k
)  x.  ( 2 ^ ( k  +  1 ) ) ) )
1981482timesd 10559 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2  x.  k )  =  ( k  +  k ) )
199198oveq1d 6101 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  =  ( ( k  +  k )  +  1 ) )
200148, 148, 169addassd 9400 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( k  +  k )  +  1 )  =  ( k  +  ( k  +  1 ) ) )
201199, 200eqtrd 2469 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2  x.  k )  +  1 )  =  ( k  +  ( k  +  1 ) ) )
202201oveq2d 6102 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( ( 2  x.  k )  +  1 ) )  =  ( 2 ^ ( k  +  ( k  +  1 ) ) ) )
203102nnrpd 11018 . . . . . . . . . 10  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( 2 ^ ( k  +  1 ) )  e.  RR+ )
204203rprege0d 11026 . . . . . . . . 9  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ ( k  +  1 ) )  e.  RR  /\  0  <_ 
( 2 ^ (
k  +  1 ) ) ) )
205 sqrsq 12751 . . . . . . . . 9  |-  ( ( ( 2 ^ (
k  +  1 ) )  e.  RR  /\  0  <_  ( 2 ^ ( k  +  1 ) ) )  -> 
( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  =  ( 2 ^ ( k  +  1 ) ) )
206204, 205syl 16 . . . . . . . 8  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) )  =  ( 2 ^ ( k  +  1 ) ) )
207206oveq2d 6102 . . . . . . 7  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ k )  x.  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) ) )  =  ( ( 2 ^ k
)  x.  ( 2 ^ ( k  +  1 ) ) ) )
208197, 202, 2073eqtr4rd 2480 . . . . . 6  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  ( ( 2 ^ k )  x.  ( sqr `  (
( 2 ^ (
k  +  1 ) ) ^ 2 ) ) )  =  ( 2 ^ ( ( 2  x.  k )  +  1 ) ) )
209194, 208breq12d 4298 . . . . 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 ) ) ) )
210122, 166, 2093imtr3d 267 . . . 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 ) ) ) )
21196, 210mtod 177 . . 3  |-  ( (  seq 1 (  +  ,  F )  e. 
dom 
~~>  /\  k  e.  NN )  ->  -.  ( sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  -  ( 1  / 
2 ) )  < 
(  seq 1 (  +  ,  F ) `  k ) )
212211nrexdv 2813 . 2  |-  (  seq 1 (  +  ,  F )  e.  dom  ~~>  ->  -.  E. k  e.  NN  ( sup ( ran  seq 1 (  +  ,  F ) ,  RR ,  <  )  -  (
1  /  2 ) )  <  (  seq 1 (  +  ,  F ) `  k
) )
21387, 212pm2.65i 173 1  |-  -.  seq 1 (  +  ,  F )  e.  dom  ~~>
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2600   A.wral 2709   E.wrex 2710   {crab 2713    \ cdif 3318    C_ wss 3321   (/)c0 3630   ifcif 3784   class class class wbr 4285    e. cmpt 4343   dom cdm 4832   ran crn 4833    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   2c2 10363   NN0cn0 10571   ZZ>=cuz 10853   RR+crp 10983   ...cfz 11429    seqcseq 11798   ^cexp 11857   sqrcsqr 12714    ~~> cli 12954   sum_csu 13155    || cdivides 13527   Primecprime 13755
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  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
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  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-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  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-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896
This theorem is referenced by:  prmrec  13975
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