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Theorem chpchtsum 23692
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 12065 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
2 inss2 3705 . . . . . . . . . 10  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
3 simpr 459 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
42, 3sseldi 3487 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
5 prmnn 14304 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
64, 5syl 16 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
76nnrpd 11257 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
87relogcld 23176 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
98recnd 9611 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
10 fsumconst 13687 . . . . 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 659 . . . 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 455 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
13 1red 9600 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  e.  RR )
146nnred 10546 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
15 prmuz2 14319 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
164, 15syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ZZ>= ` 
2 ) )
17 eluz2b2 11155 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
1817simprbi 462 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
1916, 18syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  p )
20 inss1 3704 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
2120, 3sseldi 3487 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
22 0re 9585 . . . . . . . . . . . . . 14  |-  0  e.  RR
23 elicc2 11592 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2422, 12, 23sylancr 661 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2521, 24mpbid 210 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
2625simp3d 1008 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
2713, 14, 12, 19, 26ltletrd 9731 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
2812, 27rplogcld 23182 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
2914, 19rplogcld 23182 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
3028, 29rpdivcld 11276 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
3130rpred 11259 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
3230rpge0d 11263 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
33 flge0nn0 11936 . . . . . . 7  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
3431, 32, 33syl2anc 659 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
35 hashfz1 12401 . . . . . 6  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3634, 35syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3736oveq1d 6285 . . . 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 11916 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
3938zcnd 10966 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
4039, 9mulcomd 9606 . . . 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 2500 . . 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 13607 . 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 23691 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
44 simpl 455 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
45 0red 9586 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  e.  RR )
46 1red 9600 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  e.  RR )
47 0lt1 10071 . . . . . . . . 9  |-  0  <  1
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  1
)
49 elfzuz2 11694 . . . . . . . . 9  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
50 eluzle 11094 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  <_  ( |_ `  A ) )
5150adantl 464 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  ( |_ `  A
) )
52 simpl 455 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  A  e.  RR )
53 1z 10890 . . . . . . . . . . 11  |-  1  e.  ZZ
54 flge 11923 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( 1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5552, 53, 54sylancl 660 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  (
1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5651, 55mpbird 232 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  A )
5749, 56sylan2 472 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  <_  A
)
5845, 46, 44, 48, 57ltletrd 9731 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  A
)
5945, 44, 58ltled 9722 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  A
)
60 elfznn 11717 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
6160adantl 464 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
6261nnrecred 10577 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  / 
k )  e.  RR )
6344, 59, 62recxpcld 23272 . . . . 5  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  ^c  ( 1  / 
k ) )  e.  RR )
64 chtval 23582 . . . . 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 16 . . . 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 13607 . . 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 23577 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
68 fzfid 12065 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
692sseli 3485 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
70 elfznn 11717 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
7169, 70anim12i 564 . . . . . . 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 9586 . . . . . . . . 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 3489 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7675, 5syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
7776nnred 10546 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
7876nngt0d 10575 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  p )
7973, 77, 12, 78, 26ltletrd 9731 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  A )
8079ex 432 . . . . . . 7  |-  ( A  e.  RR  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  ->  0  <  A ) )
8180adantrd 466 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  0  <  A ) )
8272, 81jcad 531 . . . . 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 3705 . . . . . . . . 9  |-  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )  C_ 
Prime
8483sseli 3485 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  ->  p  e.  Prime )
8560, 84anim12ci 565 . . . . . . 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 432 . . . . . . 7  |-  ( A  e.  RR  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  -> 
0  <  A )
)
8887adantrd 466 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  0  <  A ) )
8986, 88jcad 531 . . . . 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 3673 . . . . . . . . 9  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
91 simprll 761 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  Prime )
9291biantrud 505 . . . . . . . . . 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 9586 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  e.  RR )
94 simpl 455 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR )
9591, 5syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN )
9695nnred 10546 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR )
9795nnnn0d 10848 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN0 )
9897nn0ge0d 10851 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  p )
99 df-3an 973 . . . . . . . . . . . . 13  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
10023, 99syl6bb 261 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
101100baibd 907 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
10293, 94, 96, 98, 101syl22anc 1227 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
10392, 102bitr3d 255 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] A )  /\  p  e.  Prime )  <-> 
p  <_  A )
)
10490, 103syl5bb 257 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  <_  A ) )
105 simprr 755 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <  A )
10694, 105elrpd 11256 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR+ )
107106relogcld 23176 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  A )  e.  RR )
10891, 15syl 16 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  ( ZZ>= `  2 )
)
109108, 18syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <  p )
11096, 109rplogcld 23182 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  p )  e.  RR+ )
111107, 110rerpdivcld 11286 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
112 simprlr 762 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN )
113112nnzd 10964 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ZZ )
114 flge 11923 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
115111, 113, 114syl2anc 659 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
116112nnnn0d 10848 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN0 )
11795, 116nnexpcld 12313 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  NN )
118117nnrpd 11257 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR+ )
119118, 106logled 23180 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( log `  ( p ^ k
) )  <_  ( log `  A ) ) )
12095nnrpd 11257 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR+ )
121 relogexp 23149 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
122120, 113, 121syl2anc 659 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
123122breq1d 4449 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  (
p ^ k ) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
124112nnred 10546 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR )
125124, 107, 110lemuldivd 11304 . . . . . . . . . 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 279 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
127 nnuz 11117 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
128112, 127syl6eleq 2552 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ( ZZ>= `  1 )
)
129111flcld 11916 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )
130 elfz5 11683 . . . . . . . . . 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 659 . . . . . . . . 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 286 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  ( p ^
k )  <_  A
) )
133104, 132anbi12d 708 . . . . . . 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 11916 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  A )  e.  ZZ )
135 elfz5 11683 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
136128, 134, 135syl2anc 659 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
137 flge 11923 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  ZZ )  ->  ( k  <_  A  <->  k  <_  ( |_ `  A ) ) )
13894, 113, 137syl2anc 659 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  A  <->  k  <_  ( |_ `  A ) ) )
139136, 138bitr4d 256 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  A ) )
140 elin 3673 . . . . . . . . . 10  |-  ( p  e.  ( ( 0 [,] ( A  ^c  ( 1  / 
k ) ) )  i^i  Prime )  <->  ( p  e.  ( 0 [,] ( A  ^c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) )
14191biantrud 505 . . . . . . . . . . . 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 11263 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  A )
143112nnrecred 10577 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
1  /  k )  e.  RR )
14494, 142, 143recxpcld 23272 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( 1  /  k ) )  e.  RR )
145 elicc2 11592 . . . . . . . . . . . . . . 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 973 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^c 
( 1  /  k
) ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^c  ( 1  / 
k ) ) ) )
147145, 146syl6bb 261 . . . . . . . . . . . . . 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 907 . . . . . . . . . . . . 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 1227 . . . . . . . . . . . 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 255 . . . . . . . . . . 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 23273 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  ( A  ^c 
( 1  /  k
) ) )
152112nnrpd 11257 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR+ )
15396, 98, 144, 151, 152cxple2d 23276 . . . . . . . . . . 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 10547 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  CC )
155 cxpexp 23217 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  k  e.  NN0 )  -> 
( p  ^c 
k )  =  ( p ^ k ) )
156154, 116, 155syl2anc 659 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  ^c  k )  =  ( p ^ k ) )
157112nncnd 10547 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  CC )
158112nnne0d 10576 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  =/=  0 )
159157, 158recid2d 10312 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( 1  /  k
)  x.  k )  =  1 )
160159oveq2d 6286 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  ( ( 1  /  k )  x.  k ) )  =  ( A  ^c  1 ) )
161106, 143, 157cxpmuld 23283 . . . . . . . . . . . . 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 9611 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  CC )
163162cxp1d 23255 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^c  1 )  =  A )
164160, 161, 1633eqtr3d 2503 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( A  ^c 
( 1  /  k
) )  ^c 
k )  =  A )
165156, 164breq12d 4452 . . . . . . . . . . 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 279 . . . . . . . . . 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 257 . . . . . . . . 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 708 . . . . . . . 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 10546 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR )
170 bernneq3 12276 . . . . . . . . . . . 12  |-  ( ( p  e.  ( ZZ>= ` 
2 )  /\  k  e.  NN0 )  ->  k  <  ( p ^ k
) )
171108, 116, 170syl2anc 659 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <  ( p ^ k
) )
172124, 169, 171ltled 9722 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <_  ( p ^ k
) )
173 letr 9667 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( k  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  k  <_  A ) )
174124, 169, 94, 173syl3anc 1226 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  k  <_  A ) )
175172, 174mpand 673 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  k  <_  A ) )
176175pm4.71rd 633 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( k  <_  A  /\  ( p ^ k )  <_  A ) ) )
177154exp1d 12287 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  =  p )
17895nnge1d 10574 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <_  p )
17996, 178, 128leexp2ad 12324 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
180177, 179eqbrtrrd 4461 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  <_  ( p ^ k
) )
181 letr 9667 . . . . . . . . . . 11  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
18296, 169, 94, 181syl3anc 1226 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
183180, 182mpand 673 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
184183pm4.71rd 633 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
185168, 176, 1843bitr2rd 282 . . . . . . 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 253 . . . . . 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 432 . . . . 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 352 . . . 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 714 . . . 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 13671 . . 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 2498 . 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 2505 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    i^i cin 3460    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617    <_ cle 9618    / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   [,]cicc 11535   ...cfz 11675   |_cfl 11908   ^cexp 12148   #chash 12387   sum_csu 13590   Primecprime 14301   logclog 23108    ^c ccxp 23109   thetaccht 23562  ψcchp 23564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111  df-cht 23568  df-vma 23569  df-chp 23570
This theorem is referenced by: (None)
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