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Theorem chpchtsum 20460
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 11037 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  Fin )
2 inss2 3392 . . . . . . . . . 10  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
3 simpr 447 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
42, 3sseldi 3180 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
5 prmnn 12763 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
64, 5syl 15 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
76nnrpd 10391 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
87relogcld 19976 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
98recnd 8863 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  CC )
10 fsumconst 12254 . . . . 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 642 . . . 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 443 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
13 1re 8839 . . . . . . . . . . . 12  |-  1  e.  RR
1413a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  e.  RR )
156nnred 9763 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
16 prmuz2 12778 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
174, 16syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ZZ>= ` 
2 ) )
18 eluz2b2 10292 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
1918simprbi 450 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2017, 19syl 15 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  p )
21 inss1 3391 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
2221, 3sseldi 3180 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
23 0re 8840 . . . . . . . . . . . . . 14  |-  0  e.  RR
24 elicc2 10717 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2523, 12, 24sylancr 644 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
2622, 25mpbid 201 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
2726simp3d 969 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
2814, 15, 12, 20, 27ltletrd 8978 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
1  <  A )
2912, 28rplogcld 19982 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR+ )
3015, 20rplogcld 19982 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
3129, 30rpdivcld 10409 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR+ )
3231rpred 10392 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
3331rpge0d 10396 . . . . . . 7  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <_  ( ( log `  A )  / 
( log `  p
) ) )
34 flge0nn0 10950 . . . . . . 7  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  0  <_ 
( ( log `  A
)  /  ( log `  p ) ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
3532, 33, 34syl2anc 642 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  NN0 )
36 hashfz1 11347 . . . . . 6  |-  ( ( |_ `  ( ( log `  A )  /  ( log `  p
) ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3735, 36syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( # `  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  =  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) )
3837oveq1d 5875 . . . 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 10932 . . . . . 6  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
4039zcnd 10120 . . . . 5  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  CC )
4140, 9mulcomd 8858 . . . 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 2322 . . 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 12178 . 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 20459 . 2  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
45 simpl 443 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
4623a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  e.  RR )
4713a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  e.  RR )
48 0lt1 9298 . . . . . . . . 9  |-  0  <  1
4948a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  1
)
50 elfzuz2 10803 . . . . . . . . 9  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  ( |_ `  A )  e.  ( ZZ>= `  1 )
)
51 eluzle 10242 . . . . . . . . . . 11  |-  ( ( |_ `  A )  e.  ( ZZ>= `  1
)  ->  1  <_  ( |_ `  A ) )
5251adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  ( |_ `  A
) )
53 simpl 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  A  e.  RR )
54 1z 10055 . . . . . . . . . . 11  |-  1  e.  ZZ
55 flge 10939 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  e.  ZZ )  ->  ( 1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5653, 54, 55sylancl 643 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  (
1  <_  A  <->  1  <_  ( |_ `  A ) ) )
5752, 56mpbird 223 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( |_ `  A )  e.  ( ZZ>= `  1
) )  ->  1  <_  A )
5850, 57sylan2 460 . . . . . . . 8  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  1  <_  A
)
5946, 47, 45, 49, 58ltletrd 8978 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <  A
)
6046, 45, 59ltled 8969 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  0  <_  A
)
61 elfznn 10821 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
6261adantl 452 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
6362nnrecred 9793 . . . . . 6  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1  / 
k )  e.  RR )
6445, 60, 63recxpcld 20072 . . . . 5  |-  ( ( A  e.  RR  /\  k  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  ^ c  ( 1  / 
k ) )  e.  RR )
65 chtval 20350 . . . . 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 15 . . . 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 12178 . . 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 20345 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
69 fzfid 11037 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
702sseli 3178 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
71 elfznn 10821 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
7270, 71anim12i 549 . . . . . . 7  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
7372a1i 10 . . . . . 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 10 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  e.  RR )
752a1i 10 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  Prime )
7675sselda 3182 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7776, 5syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
7877nnred 9763 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  p  e.  RR )
7977nngt0d 9791 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  p )
8074, 78, 12, 79, 27ltletrd 8978 . . . . . . . 8  |-  ( ( A  e.  RR  /\  p  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
0  <  A )
8180ex 423 . . . . . . 7  |-  ( A  e.  RR  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  ->  0  <  A ) )
8281adantrd 454 . . . . . 6  |-  ( A  e.  RR  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  ->  0  <  A ) )
8373, 82jcad 519 . . . . 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 3392 . . . . . . . . 9  |-  ( ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )  C_ 
Prime
8584sseli 3178 . . . . . . . 8  |-  ( p  e.  ( ( 0 [,] ( A  ^ c  ( 1  / 
k ) ) )  i^i  Prime )  ->  p  e.  Prime )
8661, 85anim12ci 550 . . . . . . 7  |-  ( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  p  e.  ( (
0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  ( p  e.  Prime  /\  k  e.  NN ) )
8786a1i 10 . . . . . 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 423 . . . . . . 7  |-  ( A  e.  RR  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  -> 
0  <  A )
)
8988adantrd 454 . . . . . 6  |-  ( A  e.  RR  ->  (
( k  e.  ( 1 ... ( |_
`  A ) )  /\  p  e.  ( ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  i^i  Prime )
)  ->  0  <  A ) )
9087, 89jcad 519 . . . . 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 3360 . . . . . . . . 9  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
92 simprll 738 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  Prime )
9392biantrud 493 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) ) )
9423a1i 10 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  e.  RR )
95 simpl 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR )
9692, 5syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN )
9796nnred 9763 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR )
9896nnnn0d 10020 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  NN0 )
9998nn0ge0d 10023 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  p )
100 df-3an 936 . . . . . . . . . . . . 13  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
10124, 100syl6bb 252 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
102101baibd 875 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
10394, 95, 97, 99, 102syl22anc 1183 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
10493, 103bitr3d 246 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  e.  ( 0 [,] A )  /\  p  e.  Prime )  <-> 
p  <_  A )
)
10591, 104syl5bb 248 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  <_  A ) )
106 simprr 733 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <  A )
10795, 106elrpd 10390 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  RR+ )
108107relogcld 19976 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  A )  e.  RR )
10992, 16syl 15 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  ( ZZ>= `  2 )
)
110109, 19syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <  p )
11197, 110rplogcld 19982 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  p )  e.  RR+ )
112108, 111rerpdivcld 10419 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
113 simprlr 739 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN )
114113nnzd 10118 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ZZ )
115 flge 10939 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
116112, 114, 115syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
117113nnnn0d 10020 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  NN0 )
11896, 117nnexpcld 11268 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  NN )
119118nnrpd 10391 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR+ )
120119, 107logled 19980 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( log `  ( p ^ k
) )  <_  ( log `  A ) ) )
12196nnrpd 10391 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  RR+ )
122 relogexp 19951 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
123121, 114, 122syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
124123breq1d 4035 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( log `  (
p ^ k ) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
125113nnred 9763 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR )
126125, 108, 111lemuldivd 10437 . . . . . . . . . 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 270 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
128 nnuz 10265 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
129113, 128syl6eleq 2375 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  ( ZZ>= `  1 )
)
130112flcld 10932 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )
131 elfz5 10792 . . . . . . . . . 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 642 . . . . . . . . 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 277 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  <->  ( p ^
k )  <_  A
) )
134105, 133anbi12d 691 . . . . . . 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 10932 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( |_ `  A )  e.  ZZ )
136 elfz5 10792 . . . . . . . . . . 11  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
137129, 135, 136syl2anc 642 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  ( |_ `  A ) ) )
138 flge 10939 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  k  e.  ZZ )  ->  ( k  <_  A  <->  k  <_  ( |_ `  A ) ) )
13995, 114, 138syl2anc 642 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  <_  A  <->  k  <_  ( |_ `  A ) ) )
140137, 139bitr4d 247 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  k  <_  A ) )
141 elin 3360 . . . . . . . . . 10  |-  ( p  e.  ( ( 0 [,] ( A  ^ c  ( 1  / 
k ) ) )  i^i  Prime )  <->  ( p  e.  ( 0 [,] ( A  ^ c  ( 1  /  k ) ) )  /\  p  e. 
Prime ) )
14292biantrud 493 . . . . . . . . . . . 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 10396 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  A )
144113nnrecred 9793 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
1  /  k )  e.  RR )
14595, 143, 144recxpcld 20072 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  ( 1  /  k ) )  e.  RR )
146 elicc2 10717 . . . . . . . . . . . . . . 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 936 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( A  ^ c 
( 1  /  k
) ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( A  ^ c  ( 1  / 
k ) ) ) )
148146, 147syl6bb 252 . . . . . . . . . . . . . 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 875 . . . . . . . . . . . . 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 1183 . . . . . . . . . . . 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 246 . . . . . . . . . . 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 20073 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  0  <_  ( A  ^ c 
( 1  /  k
) ) )
153113nnrpd 10391 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  RR+ )
15497, 99, 145, 152, 153cxple2d 20076 . . . . . . . . . . 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 9764 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  e.  CC )
156 cxpexp 20017 . . . . . . . . . . . . 13  |-  ( ( p  e.  CC  /\  k  e.  NN0 )  -> 
( p  ^ c 
k )  =  ( p ^ k ) )
157155, 117, 156syl2anc 642 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p  ^ c  k )  =  ( p ^ k ) )
158113nncnd 9764 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  e.  CC )
159113nnne0d 9792 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  =/=  0 )
160158, 159recid2d 9534 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( 1  /  k
)  x.  k )  =  1 )
161160oveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  ( ( 1  /  k )  x.  k ) )  =  ( A  ^ c  1 ) )
162107, 144, 158cxpmuld 20083 . . . . . . . . . . . . 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 8863 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  A  e.  CC )
164163cxp1d 20055 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  ( A  ^ c  1 )  =  A )
165161, 162, 1643eqtr3d 2325 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( A  ^ c 
( 1  /  k
) )  ^ c 
k )  =  A )
166157, 165breq12d 4038 . . . . . . . . . . 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 270 . . . . . . . . . 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 248 . . . . . . . . 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 691 . . . . . . . 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 9763 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ k )  e.  RR )
171 bernneq3 11231 . . . . . . . . . . . 12  |-  ( ( p  e.  ( ZZ>= ` 
2 )  /\  k  e.  NN0 )  ->  k  <  ( p ^ k
) )
172109, 117, 171syl2anc 642 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <  ( p ^ k
) )
173125, 170, 172ltled 8969 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  k  <_  ( p ^ k
) )
174 letr 8916 . . . . . . . . . . 11  |-  ( ( k  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( k  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  k  <_  A ) )
175125, 170, 95, 174syl3anc 1182 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( k  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  k  <_  A ) )
176173, 175mpand 656 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  k  <_  A ) )
177176pm4.71rd 616 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( k  <_  A  /\  ( p ^ k )  <_  A ) ) )
178155exp1d 11242 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  =  p )
17996nnge1d 9790 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  1  <_  p )
18097, 179, 129leexp2ad 11279 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
181178, 180eqbrtrrd 4047 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  p  <_  ( p ^ k
) )
182 letr 8916 . . . . . . . . . . 11  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
18397, 170, 95, 182syl3anc 1182 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
184181, 183mpand 656 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
185184pm4.71rd 616 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( ( p  e. 
Prime  /\  k  e.  NN )  /\  0  <  A
) )  ->  (
( p ^ k
)  <_  A  <->  ( p  <_  A  /\  ( p ^ k )  <_  A ) ) )
186169, 177, 1853bitr2rd 273 . . . . . . 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 244 . . . . . 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 423 . . . . 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 343 . . . 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 697 . . . 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 12239 . . 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 2320 . 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 2327 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 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Fincfn 6865   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   2c2 9797   NN0cn0 9967   ZZcz 10026   ZZ>=cuz 10232   RR+crp 10356   [,]cicc 10661   ...cfz 10784   |_cfl 10926   ^cexp 11106   #chash 11339   sum_csu 12160   Primecprime 12760   logclog 19914    ^ c ccxp 19915   thetaccht 20330  ψcchp 20332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917  df-cht 20336  df-vma 20337  df-chp 20338
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