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Theorem chpchtsum 20956
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 11267 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
2 inss2 3522 . . . . . . . . . 10  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
3 simpr 448 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
42, 3sseldi 3306 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
5 prmnn 13037 . . . . . . . . 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 10603 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
87relogcld 20471 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
98recnd 9070 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
10 fsumconst 12528 . . . . 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 643 . . . 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 444 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
13 1re 9046 . . . . . . . . . . . 12  |-  1  e.  RR
1413a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  e.  RR )
156nnred 9971 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
16 prmuz2 13052 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
174, 16syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ZZ>= ` 
2 ) )
18 eluz2b2 10504 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
1918simprbi 451 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2017, 19syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  p )
21 inss1 3521 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
2221, 3sseldi 3306 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
23 0re 9047 . . . . . . . . . . . . . 14  |-  0  e.  RR
24 elicc2 10931 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2523, 12, 24sylancr 645 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2622, 25mpbid 202 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
2726simp3d 971 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
2814, 15, 12, 20, 27ltletrd 9186 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
2912, 28rplogcld 20477 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
3015, 20rplogcld 20477 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
3129, 30rpdivcld 10621 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
3231rpred 10604 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
3331rpge0d 10608 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
34 flge0nn0 11180 . . . . . . 7  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
3532, 33, 34syl2anc 643 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
36 hashfz1 11585 . . . . . 6  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3735, 36syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3837oveq1d 6055 . . . 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 ) ) )
3932flcld 11162 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
4039zcnd 10332 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
4140, 9mulcomd 9065 . . . 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 ) ) ) ) )
4211, 38, 413eqtrrd 2441 . . 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
) )
4342sumeq2dv 12452 . 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 ) )
44 chpval2 20955 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
45 simpl 444 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
4623a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  e.  RR )
4713a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  e.  RR )
48 0lt1 9506 . . . . . . . . 9  |-  0  <  1
4948a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  1
)
50 elfzuz2 11018 . . . . . . . . 9  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
51 eluzle 10454 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  <_  ( |_ `  A ) )
5251adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  ( |_ `  A
) )
53 simpl 444 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  A  e.  RR )
54 1z 10267 . . . . . . . . . . 11  |-  1  e.  ZZ
55 flge 11169 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( 1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5653, 54, 55sylancl 644 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  (
1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5752, 56mpbird 224 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  A )
5850, 57sylan2 461 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  <_  A
)
5946, 47, 45, 49, 58ltletrd 9186 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  A
)
6046, 45, 59ltled 9177 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  A
)
61 elfznn 11036 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
6261adantl 453 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
6362nnrecred 10001 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  / 
k )  e.  RR )
6445, 60, 63recxpcld 20567 . . . . 5  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  ^ c  ( 1  / 
k ) )  e.  RR )
65 chtval 20846 . . . . 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
) )
6664, 65syl 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
) )
6766sumeq2dv 12452 . . 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
) )
68 ppifi 20841 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
69 fzfid 11267 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
702sseli 3304 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
71 elfznn 11036 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
7270, 71anim12i 550 . . . . . . 7  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
7372a1i 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 ) ) )
7423a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  e.  RR )
752a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  Prime )
7675sselda 3308 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7776, 5syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
7877nnred 9971 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
7977nngt0d 9999 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  p )
8074, 78, 12, 79, 27ltletrd 9186 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  A )
8180ex 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  ->  0  <  A ) )
8281adantrd 455 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  0  <  A ) )
8373, 82jcad 520 . . . . 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
) ) )
84 inss2 3522 . . . . . . . . 9  |-  ( ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )  C_ 
Prime
8584sseli 3304 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] ( A  ^ c  ( 1  / 
k ) ) )  i^i  Prime )  ->  p  e.  Prime )
8661, 85anim12ci 551 . . . . . . 7  |-  ( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( (
0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  ( p  e.  Prime  /\  k  e.  NN ) )
8786a1i 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 ) ) )
8859ex 424 . . . . . . 7  |-  ( A  e.  RR  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  -> 
0  <  A )
)
8988adantrd 455 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  0  <  A ) )
9087, 89jcad 520 . . . . 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 ) ) )
91 elin 3490 . . . . . . . . 9  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
92 simprll 739 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  Prime )
9392biantrud 494 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) ) )
9423a1i 11 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  e.  RR )
95 simpl 444 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR )
9692, 5syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN )
9796nnred 9971 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR )
9896nnnn0d 10230 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN0 )
9998nn0ge0d 10233 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  p )
100 df-3an 938 . . . . . . . . . . . . 13  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
10124, 100syl6bb 253 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
102101baibd 876 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
10394, 95, 97, 99, 102syl22anc 1185 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
10493, 103bitr3d 247 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] A )  /\  p  e.  Prime )  <-> 
p  <_  A )
)
10591, 104syl5bb 249 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  <_  A ) )
106 simprr 734 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <  A )
10795, 106elrpd 10602 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR+ )
108107relogcld 20471 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  A )  e.  RR )
10992, 16syl 16 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  ( ZZ>= `  2 )
)
110109, 19syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <  p )
11197, 110rplogcld 20477 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  p )  e.  RR+ )
112108, 111rerpdivcld 10631 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
113 simprlr 740 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN )
114113nnzd 10330 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ZZ )
115 flge 11169 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
116112, 114, 115syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
117113nnnn0d 10230 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN0 )
11896, 117nnexpcld 11499 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  NN )
119118nnrpd 10603 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR+ )
120119, 107logled 20475 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( log `  ( p ^ k
) )  <_  ( log `  A ) ) )
12196nnrpd 10603 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR+ )
122 relogexp 20443 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
123121, 114, 122syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
124123breq1d 4182 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  (
p ^ k ) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
125113nnred 9971 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR )
126125, 108, 111lemuldivd 10649 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  x.  ( log `  p ) )  <_  ( log `  A
)  <->  k  <_  (
( log `  A
)  /  ( log `  p ) ) ) )
127120, 124, 1263bitrd 271 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
128 nnuz 10477 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
129113, 128syl6eleq 2494 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ( ZZ>= `  1 )
)
130112flcld 11162 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )
131 elfz5 11007 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
132129, 130, 131syl2anc 643 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
133116, 127, 1323bitr4rd 278 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  ( p ^
k )  <_  A
) )
134105, 133anbi12d 692 . . . . . . 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 ) ) )
13595flcld 11162 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  A )  e.  ZZ )
136 elfz5 11007 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
137129, 135, 136syl2anc 643 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
138 flge 11169 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  ZZ )  ->  ( k  <_  A  <->  k  <_  ( |_ `  A ) ) )
13995, 114, 138syl2anc 643 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  A  <->  k  <_  ( |_ `  A ) ) )
140137, 139bitr4d 248 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  A ) )
141 elin 3490 . . . . . . . . . 10  |-  ( p  e.  ( ( 0 [,] ( A  ^ c  ( 1  / 
k ) ) )  i^i  Prime )  <->  ( p  e.  ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) )
14292biantrud 494 . . . . . . . . . . . 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 ) ) )
143107rpge0d 10608 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  A )
144113nnrecred 10001 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
1  /  k )  e.  RR )
14595, 143, 144recxpcld 20567 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  ( 1  /  k ) )  e.  RR )
146 elicc2 10931 . . . . . . . . . . . . . . 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 ) ) ) ) )
147 df-3an 938 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^ c 
( 1  /  k
) ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^ c  ( 1  / 
k ) ) ) )
148146, 147syl6bb 253 . . . . . . . . . . . . . 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
) ) ) ) )
149148baibd 876 . . . . . . . . . . . . 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 ) ) ) )
15094, 145, 97, 99, 149syl22anc 1185 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] ( A  ^ c  ( 1  / 
k ) ) )  <-> 
p  <_  ( A  ^ c  ( 1  /  k ) ) ) )
151142, 150bitr3d 247 . . . . . . . . . . 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 ) ) ) )
15295, 143, 144cxpge0d 20568 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  ( A  ^ c 
( 1  /  k
) ) )
153113nnrpd 10603 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR+ )
15497, 99, 145, 152, 153cxple2d 20571 . . . . . . . . . . 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 ) ) )
15596nncnd 9972 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  CC )
156 cxpexp 20512 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  k  e.  NN0 )  -> 
( p  ^ c 
k )  =  ( p ^ k ) )
157155, 117, 156syl2anc 643 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  ^ c  k )  =  ( p ^ k ) )
158113nncnd 9972 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  CC )
159113nnne0d 10000 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  =/=  0 )
160158, 159recid2d 9742 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( 1  /  k
)  x.  k )  =  1 )
161160oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  ( ( 1  /  k )  x.  k ) )  =  ( A  ^ c  1 ) )
162107, 144, 158cxpmuld 20578 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  ( ( 1  /  k )  x.  k ) )  =  ( ( A  ^ c  ( 1  /  k ) )  ^ c  k ) )
16395recnd 9070 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  CC )
164163cxp1d 20550 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  1 )  =  A )
165161, 162, 1643eqtr3d 2444 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( A  ^ c 
( 1  /  k
) )  ^ c 
k )  =  A )
166157, 165breq12d 4185 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  ^ c 
k )  <_  (
( A  ^ c 
( 1  /  k
) )  ^ c 
k )  <->  ( p ^ k )  <_  A ) )
167151, 154, 1663bitrd 271 . . . . . . . . . 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
) )
168141, 167syl5bb 249 . . . . . . . . 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 ) )
169140, 168anbi12d 692 . . . . . . . 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
) ) )
170118nnred 9971 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR )
171 bernneq3 11462 . . . . . . . . . . . 12  |-  ( ( p  e.  ( ZZ>= ` 
2 )  /\  k  e.  NN0 )  ->  k  <  ( p ^ k
) )
172109, 117, 171syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <  ( p ^ k
) )
173125, 170, 172ltled 9177 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <_  ( p ^ k
) )
174 letr 9123 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( k  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  k  <_  A ) )
175125, 170, 95, 174syl3anc 1184 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  k  <_  A ) )
176173, 175mpand 657 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  k  <_  A ) )
177176pm4.71rd 617 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( k  <_  A  /\  ( p ^ k )  <_  A ) ) )
178155exp1d 11473 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  =  p )
17996nnge1d 9998 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <_  p )
18097, 179, 129leexp2ad 11510 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
181178, 180eqbrtrrd 4194 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  <_  ( p ^ k
) )
182 letr 9123 . . . . . . . . . . 11  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
18397, 170, 95, 182syl3anc 1184 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
184181, 183mpand 657 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
185184pm4.71rd 617 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
186169, 177, 1853bitr2rd 274 . . . . . . 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 )
) ) )
187134, 186bitrd 245 . . . . . 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 )
) ) )
188187ex 424 . . . . 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 )
) ) ) )
18983, 90, 188pm5.21ndd 344 . . . 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 )
) ) )
1909adantrr 698 . . . 4  |-  ( ( A  e.  RR  /\  ( p  e.  (
( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )  -> 
( log `  p
)  e.  CC )
19168, 69, 1, 189, 190fsumcom2 12513 . . 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 ) )
19267, 191eqtr4d 2439 . 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 ) )
19343, 44, 1923eqtr4d 2446 1  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ k  e.  ( 1 ... ( |_ `  A ) ) (
theta `  ( A  ^ c  ( 1  / 
k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   [,]cicc 10875   ...cfz 10999   |_cfl 11156   ^cexp 11337   #chash 11573   sum_csu 12434   Primecprime 13034   logclog 20405    ^ c ccxp 20406   thetaccht 20826  ψcchp 20828
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-cht 20832  df-vma 20833  df-chp 20834
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