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Theorem chpchtsum 24147
Description: The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
chpchtsum  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ k  e.  ( 1 ... ( |_ `  A ) ) (
theta `  ( A  ^c  ( 1  / 
k ) ) ) )
Distinct variable group:    A, k

Proof of Theorem chpchtsum
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 fzfid 12186 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
2 inss2 3653 . . . . . . . . . 10  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
3 simpr 463 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
42, 3sseldi 3430 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
5 prmnn 14625 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
64, 5syl 17 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
76nnrpd 11339 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
87relogcld 23572 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
98recnd 9669 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
10 fsumconst 13851 . . . . 5  |-  ( ( ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin  /\  ( log `  p
)  e.  CC )  ->  sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p )  =  ( ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) ) )
111, 9, 10syl2anc 667 . . . 4  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ( log `  p
)  =  ( (
# `  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) ) )
12 simpl 459 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
13 1red 9658 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  e.  RR )
146nnred 10624 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
15 prmuz2 14642 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
164, 15syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ZZ>= ` 
2 ) )
17 eluz2b2 11231 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
1817simprbi 466 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
1916, 18syl 17 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  p )
20 inss1 3652 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
2120, 3sseldi 3430 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
22 0re 9643 . . . . . . . . . . . . . 14  |-  0  e.  RR
23 elicc2 11699 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2422, 12, 23sylancr 669 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2521, 24mpbid 214 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
2625simp3d 1022 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
2713, 14, 12, 19, 26ltletrd 9795 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
2812, 27rplogcld 23578 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
2914, 19rplogcld 23578 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
3028, 29rpdivcld 11358 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
3130rpred 11341 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
3230rpge0d 11345 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
33 flge0nn0 12054 . . . . . . 7  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
3431, 32, 33syl2anc 667 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
35 hashfz1 12529 . . . . . 6  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3634, 35syl 17 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3736oveq1d 6305 . . . 4  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  x.  ( log `  p ) )  =  ( ( |_
`  ( ( log `  A )  /  ( log `  p ) ) )  x.  ( log `  p ) ) )
3831flcld 12034 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
3938zcnd 11041 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
4039, 9mulcomd 9664 . . . 4  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  x.  ( log `  p
) )  =  ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )
4111, 37, 403eqtrrd 2490 . . 3  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  =  sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ( log `  p
) )
4241sumeq2dv 13769 . 2  |-  ( A  e.  RR  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p ) )
43 chpval2 24146 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
44 simpl 459 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
45 0red 9644 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  e.  RR )
46 1red 9658 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  e.  RR )
47 0lt1 10136 . . . . . . . . 9  |-  0  <  1
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  1
)
49 elfzuz2 11804 . . . . . . . . 9  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
50 eluzle 11171 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  <_  ( |_ `  A ) )
5150adantl 468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  ( |_ `  A
) )
52 simpl 459 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  A  e.  RR )
53 1z 10967 . . . . . . . . . . 11  |-  1  e.  ZZ
54 flge 12041 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( 1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5552, 53, 54sylancl 668 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  (
1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5651, 55mpbird 236 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  A )
5749, 56sylan2 477 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  <_  A
)
5845, 46, 44, 48, 57ltletrd 9795 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  A
)
5945, 44, 58ltled 9783 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  A
)
60 elfznn 11828 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
6160adantl 468 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
6261nnrecred 10655 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  / 
k )  e.  RR )
6344, 59, 62recxpcld 23668 . . . . 5  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  ^c  ( 1  / 
k ) )  e.  RR )
64 chtval 24037 . . . . 5  |-  ( ( A  ^c  ( 1  /  k ) )  e.  RR  ->  (
theta `  ( A  ^c  ( 1  / 
k ) ) )  =  sum_ p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
( log `  p
) )
6563, 64syl 17 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( theta `  ( A  ^c  ( 1  /  k ) ) )  =  sum_ p  e.  ( ( 0 [,] ( A  ^c 
( 1  /  k
) ) )  i^i 
Prime ) ( log `  p
) )
6665sumeq2dv 13769 . . 3  |-  ( A  e.  RR  ->  sum_ k  e.  ( 1 ... ( |_ `  A ) ) ( theta `  ( A  ^c  ( 1  /  k ) ) )  =  sum_ k  e.  ( 1 ... ( |_ `  A ) )
sum_ p  e.  (
( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
( log `  p
) )
67 ppifi 24032 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
68 fzfid 12186 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
692sseli 3428 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
70 elfznn 11828 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
7169, 70anim12i 570 . . . . . . 7  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
7271a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (
p  e.  Prime  /\  k  e.  NN ) ) )
73 0red 9644 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  e.  RR )
742a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  Prime )
7574sselda 3432 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7675, 5syl 17 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
7776nnred 10624 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
7876nngt0d 10653 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  p )
7973, 77, 12, 78, 26ltletrd 9795 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  A )
8079ex 436 . . . . . . 7  |-  ( A  e.  RR  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  ->  0  <  A ) )
8180adantrd 470 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  0  <  A ) )
8272, 81jcad 536 . . . . 5  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  0  <  A
) ) )
83 inss2 3653 . . . . . . . . 9  |-  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )  C_ 
Prime
8483sseli 3428 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  ->  p  e.  Prime )
8560, 84anim12ci 571 . . . . . . 7  |-  ( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( (
0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  ( p  e.  Prime  /\  k  e.  NN ) )
8685a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  ( p  e.  Prime  /\  k  e.  NN ) ) )
8758ex 436 . . . . . . 7  |-  ( A  e.  RR  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  -> 
0  <  A )
)
8887adantrd 470 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  0  <  A ) )
8986, 88jcad 536 . . . . 5  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  0  <  A ) ) )
90 elin 3617 . . . . . . . . 9  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
91 simprll 772 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  Prime )
9291biantrud 510 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) ) )
93 0red 9644 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  e.  RR )
94 simpl 459 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR )
9591, 5syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN )
9695nnred 10624 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR )
9795nnnn0d 10925 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN0 )
9897nn0ge0d 10928 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  p )
99 df-3an 987 . . . . . . . . . . . . 13  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
10023, 99syl6bb 265 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
101100baibd 920 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
10293, 94, 96, 98, 101syl22anc 1269 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
10392, 102bitr3d 259 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] A )  /\  p  e.  Prime )  <-> 
p  <_  A )
)
10490, 103syl5bb 261 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  <_  A ) )
105 simprr 766 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <  A )
10694, 105elrpd 11338 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR+ )
107106relogcld 23572 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  A )  e.  RR )
10891, 15syl 17 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  ( ZZ>= `  2 )
)
109108, 18syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <  p )
11096, 109rplogcld 23578 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  p )  e.  RR+ )
111107, 110rerpdivcld 11369 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
112 simprlr 773 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN )
113112nnzd 11039 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ZZ )
114 flge 12041 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
115111, 113, 114syl2anc 667 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
116112nnnn0d 10925 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN0 )
11795, 116nnexpcld 12437 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  NN )
118117nnrpd 11339 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR+ )
119118, 106logled 23576 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( log `  ( p ^ k
) )  <_  ( log `  A ) ) )
12095nnrpd 11339 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR+ )
121 relogexp 23545 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
122120, 113, 121syl2anc 667 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
123122breq1d 4412 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  (
p ^ k ) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
124112nnred 10624 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR )
125124, 107, 110lemuldivd 11387 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  x.  ( log `  p ) )  <_  ( log `  A
)  <->  k  <_  (
( log `  A
)  /  ( log `  p ) ) ) )
126119, 123, 1253bitrd 283 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
127 nnuz 11194 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
128112, 127syl6eleq 2539 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ( ZZ>= `  1 )
)
129111flcld 12034 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )
130 elfz5 11792 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
131128, 129, 130syl2anc 667 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
132115, 126, 1313bitr4rd 290 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  ( p ^
k )  <_  A
) )
133104, 132anbi12d 717 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
13494flcld 12034 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  A )  e.  ZZ )
135 elfz5 11792 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
136128, 134, 135syl2anc 667 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
137 flge 12041 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  ZZ )  ->  ( k  <_  A  <->  k  <_  ( |_ `  A ) ) )
13894, 113, 137syl2anc 667 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  A  <->  k  <_  ( |_ `  A ) ) )
139136, 138bitr4d 260 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  A ) )
140 elin 3617 . . . . . . . . . 10  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  <->  ( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) )
14191biantrud 510 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  <-> 
( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) ) )
142106rpge0d 11345 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  A )
143112nnrecred 10655 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
1  /  k )  e.  RR )
14494, 142, 143recxpcld 23668 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( 1  /  k ) )  e.  RR )
145 elicc2 11699 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  ( A  ^c 
( 1  /  k
) )  e.  RR )  ->  ( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^c  ( 1  /  k ) ) ) ) )
146 df-3an 987 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^c 
( 1  /  k
) ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^c  ( 1  / 
k ) ) ) )
147145, 146syl6bb 265 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  ( A  ^c 
( 1  /  k
) )  e.  RR )  ->  ( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  <->  ( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^c 
( 1  /  k
) ) ) ) )
148147baibd 920 . . . . . . . . . . . . 13  |-  ( ( ( 0  e.  RR  /\  ( A  ^c 
( 1  /  k
) )  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  <->  p  <_  ( A  ^c  ( 1  /  k ) ) ) )
14993, 144, 96, 98, 148syl22anc 1269 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  <-> 
p  <_  ( A  ^c  ( 1  /  k ) ) ) )
150141, 149bitr3d 259 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime )  <->  p  <_  ( A  ^c  ( 1  /  k ) ) ) )
15194, 142, 143cxpge0d 23669 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  ( A  ^c 
( 1  /  k
) ) )
152112nnrpd 11339 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR+ )
15396, 98, 144, 151, 152cxple2d 23672 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  <_  ( A  ^c  ( 1  /  k ) )  <-> 
( p  ^c 
k )  <_  (
( A  ^c 
( 1  /  k
) )  ^c 
k ) ) )
15495nncnd 10625 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  CC )
155 cxpexp 23613 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  k  e.  NN0 )  -> 
( p  ^c 
k )  =  ( p ^ k ) )
156154, 116, 155syl2anc 667 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  ^c  k )  =  ( p ^ k ) )
157112nncnd 10625 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  CC )
158112nnne0d 10654 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  =/=  0 )
159157, 158recid2d 10379 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( 1  /  k
)  x.  k )  =  1 )
160159oveq2d 6306 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( ( 1  /  k )  x.  k ) )  =  ( A  ^c  1 ) )
161106, 143, 157cxpmuld 23679 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( ( 1  /  k )  x.  k ) )  =  ( ( A  ^c  ( 1  /  k ) )  ^c  k ) )
16294recnd 9669 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  CC )
163162cxp1d 23651 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  1 )  =  A )
164160, 161, 1633eqtr3d 2493 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( A  ^c 
( 1  /  k
) )  ^c 
k )  =  A )
165156, 164breq12d 4415 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  ^c 
k )  <_  (
( A  ^c 
( 1  /  k
) )  ^c 
k )  <->  ( p ^ k )  <_  A ) )
166150, 153, 1653bitrd 283 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime )  <->  ( p ^
k )  <_  A
) )
167140, 166syl5bb 261 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )  <->  ( p ^ k )  <_  A ) )
168139, 167anbi12d 717 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  <->  ( k  <_  A  /\  ( p ^
k )  <_  A
) ) )
169117nnred 10624 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR )
170 bernneq3 12400 . . . . . . . . . . . 12  |-  ( ( p  e.  ( ZZ>= ` 
2 )  /\  k  e.  NN0 )  ->  k  <  ( p ^ k
) )
171108, 116, 170syl2anc 667 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <  ( p ^ k
) )
172124, 169, 171ltled 9783 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <_  ( p ^ k
) )
173 letr 9727 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( k  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  k  <_  A ) )
174124, 169, 94, 173syl3anc 1268 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  k  <_  A ) )
175172, 174mpand 681 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  k  <_  A ) )
176175pm4.71rd 641 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( k  <_  A  /\  ( p ^ k )  <_  A ) ) )
177154exp1d 12411 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  =  p )
17895nnge1d 10652 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <_  p )
17996, 178, 128leexp2ad 12448 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
180177, 179eqbrtrrd 4425 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  <_  ( p ^ k
) )
181 letr 9727 . . . . . . . . . . 11  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
18296, 169, 94, 181syl3anc 1268 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
183180, 182mpand 681 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
184183pm4.71rd 641 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
185168, 176, 1843bitr2rd 286 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  A  /\  ( p ^ k
)  <_  A )  <->  ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( (
0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
) ) )
186133, 185bitrd 257 . . . . . 6  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
) ) )
187186ex 436 . . . . 5  |-  ( A  e.  RR  ->  (
( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
)  ->  ( (
p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  <->  ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
) ) ) )
18882, 89, 187pm5.21ndd 356 . . . 4  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
) ) )
1899adantrr 723 . . . 4  |-  ( ( A  e.  RR  /\  ( p  e.  (
( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )  -> 
( log `  p
)  e.  CC )
19067, 68, 1, 188, 189fsumcom2 13835 . . 3  |-  ( A  e.  RR  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime )
sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p )  =  sum_ k  e.  ( 1 ... ( |_ `  A ) ) sum_ p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime ) ( log `  p ) )
19166, 190eqtr4d 2488 . 2  |-  ( A  e.  RR  ->  sum_ k  e.  ( 1 ... ( |_ `  A ) ) ( theta `  ( A  ^c  ( 1  /  k ) ) )  =  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime )
sum_ k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) ( log `  p ) )
19242, 43, 1913eqtr4d 2495 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ k  e.  ( 1 ... ( |_ `  A ) ) (
theta `  ( A  ^c  ( 1  / 
k ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Fincfn 7569   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544    < clt 9675    <_ cle 9676    / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   [,]cicc 11638   ...cfz 11784   |_cfl 12026   ^cexp 12272   #chash 12515   sum_csu 13752   Primecprime 14622   logclog 23504    ^c ccxp 23505   thetaccht 24017  ψcchp 24019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-cht 24023  df-vma 24024  df-chp 24025
This theorem is referenced by: (None)
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