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Theorem chpchtsum 24226
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 12224 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
2 inss2 3644 . . . . . . . . . 10  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
3 simpr 468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
42, 3sseldi 3416 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
5 prmnn 14704 . . . . . . . . 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 11362 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
87relogcld 23651 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
98recnd 9687 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
10 fsumconst 13928 . . . . 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 673 . . . 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 464 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
13 1red 9676 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  e.  RR )
146nnred 10646 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
15 prmuz2 14721 . . . . . . . . . . . . 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 11254 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
1817simprbi 471 . . . . . . . . . . . 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 3643 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
2120, 3sseldi 3416 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
22 0re 9661 . . . . . . . . . . . . . 14  |-  0  e.  RR
23 elicc2 11724 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2422, 12, 23sylancr 676 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2521, 24mpbid 215 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
2625simp3d 1044 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
2713, 14, 12, 19, 26ltletrd 9812 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
2812, 27rplogcld 23657 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
2914, 19rplogcld 23657 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
3028, 29rpdivcld 11381 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
3130rpred 11364 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
3230rpge0d 11368 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
33 flge0nn0 12087 . . . . . . 7  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
3431, 32, 33syl2anc 673 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
35 hashfz1 12567 . . . . . 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 6323 . . . 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 12067 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
3938zcnd 11064 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
4039, 9mulcomd 9682 . . . 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 2510 . . 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 13846 . 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 24225 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
44 simpl 464 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
45 0red 9662 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  e.  RR )
46 1red 9676 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  e.  RR )
47 0lt1 10157 . . . . . . . . 9  |-  0  <  1
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  1
)
49 elfzuz2 11830 . . . . . . . . 9  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
50 eluzle 11195 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  <_  ( |_ `  A ) )
5150adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  ( |_ `  A
) )
52 simpl 464 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  A  e.  RR )
53 1z 10991 . . . . . . . . . . 11  |-  1  e.  ZZ
54 flge 12074 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( 1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5552, 53, 54sylancl 675 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  (
1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5651, 55mpbird 240 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  A )
5749, 56sylan2 482 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  <_  A
)
5845, 46, 44, 48, 57ltletrd 9812 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  A
)
5945, 44, 58ltled 9800 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  A
)
60 elfznn 11854 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
6160adantl 473 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
6261nnrecred 10677 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  / 
k )  e.  RR )
6344, 59, 62recxpcld 23747 . . . . 5  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  ^c  ( 1  / 
k ) )  e.  RR )
64 chtval 24116 . . . . 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 13846 . . 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 24111 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
68 fzfid 12224 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
692sseli 3414 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
70 elfznn 11854 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
7169, 70anim12i 576 . . . . . . 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 9662 . . . . . . . . 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 3418 . . . . . . . . . . 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 10646 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
7876nngt0d 10675 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  p )
7973, 77, 12, 78, 26ltletrd 9812 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  A )
8079ex 441 . . . . . . 7  |-  ( A  e.  RR  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  ->  0  <  A ) )
8180adantrd 475 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  0  <  A ) )
8272, 81jcad 542 . . . . 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 3644 . . . . . . . . 9  |-  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )  C_ 
Prime
8483sseli 3414 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  ->  p  e.  Prime )
8560, 84anim12ci 577 . . . . . . 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 441 . . . . . . 7  |-  ( A  e.  RR  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  -> 
0  <  A )
)
8887adantrd 475 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  0  <  A ) )
8986, 88jcad 542 . . . . 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 3608 . . . . . . . . 9  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
91 simprll 780 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  Prime )
9291biantrud 515 . . . . . . . . . 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 9662 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  e.  RR )
94 simpl 464 . . . . . . . . . . 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 10646 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR )
9795nnnn0d 10949 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN0 )
9897nn0ge0d 10952 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  p )
99 df-3an 1009 . . . . . . . . . . . . 13  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
10023, 99syl6bb 269 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
101100baibd 923 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
10293, 94, 96, 98, 101syl22anc 1293 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
10392, 102bitr3d 263 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] A )  /\  p  e.  Prime )  <-> 
p  <_  A )
)
10490, 103syl5bb 265 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  <_  A ) )
105 simprr 774 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <  A )
10694, 105elrpd 11361 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR+ )
107106relogcld 23651 . . . . . . . . . . 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 23657 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  p )  e.  RR+ )
111107, 110rerpdivcld 11392 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
112 simprlr 781 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN )
113112nnzd 11062 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ZZ )
114 flge 12074 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
115111, 113, 114syl2anc 673 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
116112nnnn0d 10949 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN0 )
11795, 116nnexpcld 12475 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  NN )
118117nnrpd 11362 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR+ )
119118, 106logled 23655 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( log `  ( p ^ k
) )  <_  ( log `  A ) ) )
12095nnrpd 11362 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR+ )
121 relogexp 23624 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
122120, 113, 121syl2anc 673 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
123122breq1d 4405 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  (
p ^ k ) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
124112nnred 10646 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR )
125124, 107, 110lemuldivd 11410 . . . . . . . . . 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 287 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
127 nnuz 11218 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
128112, 127syl6eleq 2559 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ( ZZ>= `  1 )
)
129111flcld 12067 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )
130 elfz5 11818 . . . . . . . . . 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 673 . . . . . . . . 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 294 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  ( p ^
k )  <_  A
) )
133104, 132anbi12d 725 . . . . . . 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 12067 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  A )  e.  ZZ )
135 elfz5 11818 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
136128, 134, 135syl2anc 673 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
137 flge 12074 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  ZZ )  ->  ( k  <_  A  <->  k  <_  ( |_ `  A ) ) )
13894, 113, 137syl2anc 673 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  A  <->  k  <_  ( |_ `  A ) ) )
139136, 138bitr4d 264 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  A ) )
140 elin 3608 . . . . . . . . . 10  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  <->  ( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) )
14191biantrud 515 . . . . . . . . . . . 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 11368 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  A )
143112nnrecred 10677 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
1  /  k )  e.  RR )
14494, 142, 143recxpcld 23747 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( 1  /  k ) )  e.  RR )
145 elicc2 11724 . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^c 
( 1  /  k
) ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^c  ( 1  / 
k ) ) ) )
147145, 146syl6bb 269 . . . . . . . . . . . . . 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 923 . . . . . . . . . . . . 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 1293 . . . . . . . . . . . 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 263 . . . . . . . . . . 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 23748 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  ( A  ^c 
( 1  /  k
) ) )
152112nnrpd 11362 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR+ )
15396, 98, 144, 151, 152cxple2d 23751 . . . . . . . . . . 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 10647 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  CC )
155 cxpexp 23692 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  k  e.  NN0 )  -> 
( p  ^c 
k )  =  ( p ^ k ) )
156154, 116, 155syl2anc 673 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  ^c  k )  =  ( p ^ k ) )
157112nncnd 10647 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  CC )
158112nnne0d 10676 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  =/=  0 )
159157, 158recid2d 10401 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( 1  /  k
)  x.  k )  =  1 )
160159oveq2d 6324 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( ( 1  /  k )  x.  k ) )  =  ( A  ^c  1 ) )
161106, 143, 157cxpmuld 23758 . . . . . . . . . . . . 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 9687 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  CC )
163162cxp1d 23730 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  1 )  =  A )
164160, 161, 1633eqtr3d 2513 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( A  ^c 
( 1  /  k
) )  ^c 
k )  =  A )
165156, 164breq12d 4408 . . . . . . . . . . 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 287 . . . . . . . . . 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 265 . . . . . . . . 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 725 . . . . . . . 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 10646 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR )
170 bernneq3 12438 . . . . . . . . . . . 12  |-  ( ( p  e.  ( ZZ>= ` 
2 )  /\  k  e.  NN0 )  ->  k  <  ( p ^ k
) )
171108, 116, 170syl2anc 673 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <  ( p ^ k
) )
172124, 169, 171ltled 9800 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <_  ( p ^ k
) )
173 letr 9745 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( k  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  k  <_  A ) )
174124, 169, 94, 173syl3anc 1292 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  k  <_  A ) )
175172, 174mpand 689 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  k  <_  A ) )
176175pm4.71rd 647 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( k  <_  A  /\  ( p ^ k )  <_  A ) ) )
177154exp1d 12449 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  =  p )
17895nnge1d 10674 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <_  p )
17996, 178, 128leexp2ad 12486 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
180177, 179eqbrtrrd 4418 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  <_  ( p ^ k
) )
181 letr 9745 . . . . . . . . . . 11  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
18296, 169, 94, 181syl3anc 1292 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
183180, 182mpand 689 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
184183pm4.71rd 647 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
185168, 176, 1843bitr2rd 290 . . . . . . 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 261 . . . . . 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 441 . . . . 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 361 . . . 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 731 . . . 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 13912 . . 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 2508 . 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 2515 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   [,]cicc 11663   ...cfz 11810   |_cfl 12059   ^cexp 12310   #chash 12553   sum_csu 13829   Primecprime 14701   logclog 23583    ^c ccxp 23584   thetaccht 24096  ψcchp 24098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-cht 24102  df-vma 24103  df-chp 24104
This theorem is referenced by: (None)
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