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Theorem chpub 22562
Description: An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
chpub  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) )

Proof of Theorem chpub
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 chpcl 22465 . . . . 5  |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
21adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  e.  RR )
3 chtcl 22450 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
43adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  e.  RR )
52, 4resubcld 9779 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  e.  RR )
6 simpl 457 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR )
7 0red 9390 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  e.  RR )
8 1red 9404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  e.  RR )
9 0lt1 9865 . . . . . . . . . 10  |-  0  <  1
109a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  1 )
11 simpr 461 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  <_  A )
127, 8, 6, 10, 11ltletrd 9534 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  A )
136, 12elrpd 11028 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR+ )
1413rpge0d 11034 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  A )
156, 14resqrcld 12907 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  e.  RR )
16 ppifi 22446 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin )
1715, 16syl 16 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  e.  Fin )
1813adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  A  e.  RR+ )
1918relogcld 22075 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
2017, 19fsumrecl 13214 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  e.  RR )
2113relogcld 22075 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  RR )
2215, 21remulcld 9417 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  x.  ( log `  A ) )  e.  RR )
23 ppifi 22446 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
2423adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] A )  i^i  Prime )  e.  Fin )
25 inss2 3574 . . . . . . . . . . . 12  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
26 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A
)  i^i  Prime ) )
2725, 26sseldi 3357 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  Prime )
28 prmnn 13769 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
2927, 28syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  NN )
3029nnrpd 11029 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR+ )
3130relogcld 22075 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR )
3221adantr 465 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  A
)  e.  RR )
3329nnred 10340 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR )
34 prmuz2 13784 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
3527, 34syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  (
ZZ>= `  2 ) )
36 eluz2b2 10930 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
3736simprbi 464 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
3835, 37syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  1  <  p
)
3933, 38rplogcld 22081 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR+ )
4032, 39rerpdivcld 11057 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
41 reflcl 11649 . . . . . . . . 9  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4240, 41syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4331, 42remulcld 9417 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
4443recnd 9415 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  CC )
4531recnd 9415 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  CC )
4624, 44, 45fsumsub 13258 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  =  ( sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( log `  p
) ) )
47 0le0 10414 . . . . . . . . 9  |-  0  <_  0
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  0 )
498, 6, 6, 14, 11lemul2ad 10276 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  <_  ( A  x.  A ) )
506recnd 9415 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  CC )
5150sqsqrd 12928 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
5250mulid1d 9406 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  =  A )
5351, 52eqtr4d 2478 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  ( A  x.  1 ) )
5450sqvald 12008 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
5549, 53, 543brtr4d 4325 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  <_  ( A ^
2 ) )
566, 14sqrge0d 12910 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( sqr `  A ) )
5715, 6, 56, 14le2sqd 12046 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  <_  A  <->  ( ( sqr `  A ) ^
2 )  <_  ( A ^ 2 ) ) )
5855, 57mpbird 232 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  <_  A )
59 iccss 11366 . . . . . . . 8  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
0  /\  ( sqr `  A )  <_  A
) )  ->  (
0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
) )
607, 6, 48, 58, 59syl22anc 1219 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 0 [,] ( sqr `  A ) ) 
C_  ( 0 [,] A ) )
61 ssrin 3578 . . . . . . 7  |-  ( ( 0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
)  ->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  ( ( 0 [,] A )  i^i  Prime ) )
6260, 61syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( ( 0 [,] A )  i^i 
Prime ) )
6362sselda 3359 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6443, 31resubcld 9779 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  RR )
6564recnd 9415 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  CC )
6663, 65syldan 470 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  CC )
67 eldifi 3481 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6867, 45sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  CC )
6968mulid2d 9407 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
1  x.  ( log `  p ) )  =  ( log `  p
) )
70 inss1 3573 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
7170, 26sseldi 3357 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
72 0re 9389 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
736adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  A  e.  RR )
74 elicc2 11363 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
7572, 73, 74sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( p  e.  ( 0 [,] A
)  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) ) )
7671, 75mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) )
7776simp3d 1002 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  <_  A
)
7867, 77sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  <_  A )
7967, 30sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR+ )
8013adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR+ )
8179, 80logled 22079 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p  <_  A  <->  ( log `  p )  <_  ( log `  A ) ) )
8278, 81mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  <_ 
( log `  A
) )
8369, 82eqbrtrd 4315 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
1  x.  ( log `  p ) )  <_ 
( log `  A
) )
84 1red 9404 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  e.  RR )
8521adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  A )  e.  RR )
8667, 39sylan2 474 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  RR+ )
8784, 85, 86lemuldivd 11075 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( 1  x.  ( log `  p ) )  <_  ( log `  A
)  <->  1  <_  (
( log `  A
)  /  ( log `  p ) ) ) )
8883, 87mpbid 210 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  <_  ( ( log `  A
)  /  ( log `  p ) ) )
896adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR )
9089recnd 9415 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  CC )
9190sqsqrd 12928 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
92 eldifn 3482 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  -.  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )
9392adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  -.  p  e.  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )
9467, 27sylan2 474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  Prime )
95 elin 3542 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
( p  e.  ( 0 [,] ( sqr `  A ) )  /\  p  e.  Prime ) )
9695rbaib 898 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
p  e.  ( 0 [,] ( sqr `  A
) ) ) )
9794, 96syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  <->  p  e.  (
0 [,] ( sqr `  A ) ) ) )
98 0red 9390 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  e.  RR )
9915adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( sqr `  A )  e.  RR )
10067, 29sylan2 474 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  NN )
101100nnred 10340 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR )
10279rpge0d 11034 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  p )
103 elicc2 11363 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A ) ) ) )
104 df-3an 967 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A
) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) )
105103, 104syl6bb 261 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) ) )
106105baibd 900 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 0  e.  RR  /\  ( sqr `  A
)  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  p  <_  ( sqr `  A ) ) )
10798, 99, 101, 102, 106syl22anc 1219 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p  e.  ( 0 [,] ( sqr `  A
) )  <->  p  <_  ( sqr `  A ) ) )
10897, 107bitrd 253 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  <->  p  <_  ( sqr `  A ) ) )
10993, 108mtbid 300 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  -.  p  <_  ( sqr `  A
) )
11099, 101ltnled 9524 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
)  <  p  <->  -.  p  <_  ( sqr `  A
) ) )
111109, 110mpbird 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( sqr `  A )  < 
p )
11256adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  ( sqr `  A
) )
11399, 101, 112, 102lt2sqd 12045 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
)  <  p  <->  ( ( sqr `  A ) ^
2 )  <  (
p ^ 2 ) ) )
114111, 113mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
) ^ 2 )  <  ( p ^
2 ) )
11591, 114eqbrtrrd 4317 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  <  ( p ^ 2 ) )
116100nnsqcld 12031 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  NN )
117116nnrpd 11029 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  RR+ )
118 logltb 22051 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR+  /\  (
p ^ 2 )  e.  RR+ )  ->  ( A  <  ( p ^
2 )  <->  ( log `  A )  <  ( log `  ( p ^
2 ) ) ) )
11980, 117, 118syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( A  <  ( p ^
2 )  <->  ( log `  A )  <  ( log `  ( p ^
2 ) ) ) )
120115, 119mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  A )  < 
( log `  (
p ^ 2 ) ) )
121 2z 10681 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
122 relogexp 22047 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( p ^
2 ) )  =  ( 2  x.  ( log `  p ) ) )
12379, 121, 122sylancl 662 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  ( p ^
2 ) )  =  ( 2  x.  ( log `  p ) ) )
124120, 123breqtrd 4319 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  A )  < 
( 2  x.  ( log `  p ) ) )
125 2re 10394 . . . . . . . . . . . . . . 15  |-  2  e.  RR
126125a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  2  e.  RR )
12785, 126, 86ltdivmul2d 11078 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( ( log `  A
)  /  ( log `  p ) )  <  2  <->  ( log `  A
)  <  ( 2  x.  ( log `  p
) ) ) )
128124, 127mpbird 232 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  A
)  /  ( log `  p ) )  <  2 )
129 df-2 10383 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
130128, 129syl6breq 4334 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  A
)  /  ( log `  p ) )  < 
( 1  +  1 ) )
13167, 40sylan2 474 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  A
)  /  ( log `  p ) )  e.  RR )
132 1z 10679 . . . . . . . . . . . 12  |-  1  e.  ZZ
133 flbi 11667 . . . . . . . . . . . 12  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  1  e.  ZZ )  ->  (
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  =  1  <->  ( 1  <_  ( ( log `  A )  /  ( log `  p ) )  /\  ( ( log `  A )  /  ( log `  p ) )  <  ( 1  +  1 ) ) ) )
134131, 132, 133sylancl 662 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  =  1  <->  ( 1  <_  ( ( log `  A )  /  ( log `  p ) )  /\  ( ( log `  A )  /  ( log `  p ) )  <  ( 1  +  1 ) ) ) )
13588, 130, 134mpbir2and 913 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  =  1 )
136135oveq2d 6110 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  =  ( ( log `  p
)  x.  1 ) )
13768mulid1d 9406 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  p
)  x.  1 )  =  ( log `  p
) )
138136, 137eqtrd 2475 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  =  ( log `  p ) )
139138oveq1d 6109 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  =  ( ( log `  p )  -  ( log `  p ) ) )
14068subidd 9710 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  p
)  -  ( log `  p ) )  =  0 )
141139, 140eqtrd 2475 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  =  0 )
14262, 66, 141, 24fsumss 13205 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) ) )
143 chpval2 22560 . . . . . . 7  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
144143adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )
145 chtval 22451 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
146145adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
147144, 146oveq12d 6112 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  =  ( sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( log `  p
) ) )
14846, 142, 1473eqtr4rd 2486 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  = 
sum_ p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) ( ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) ) )
14963, 64syldan 470 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  RR )
15063, 43syldan 470 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
15163, 39syldan 470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
152151rpge0d 11034 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
0  <_  ( log `  p ) )
153 inss2 3574 . . . . . . . . . . . 12  |-  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  Prime
154 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )
155153, 154sseldi 3357 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  Prime )
156155, 28syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  NN )
157156nnrpd 11029 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  RR+ )
158157relogcld 22075 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
159150, 158subge02d 9934 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( 0  <_  ( log `  p )  <->  ( (
( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  <_  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) )
160152, 159mpbid 210 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  <_  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
16163, 40syldan 470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
162 flle 11652 . . . . . . . 8  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
163161, 162syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
16463, 42syldan 470 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  RR )
165164, 19, 151lemuldiv2d 11076 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <_  ( log `  A )  <->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) ) )
166163, 165mpbird 232 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  <_  ( log `  A ) )
167149, 150, 19, 160, 166letrd 9531 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  -  ( log `  p ) )  <_  ( log `  A
) )
16817, 149, 19, 167fsumle 13265 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
169148, 168eqbrtrd 4315 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
17021recnd 9415 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  CC )
171 fsumconst 13260 . . . . 5  |-  ( ( ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  e.  Fin  /\  ( log `  A )  e.  CC )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ( log `  A
)  =  ( (
# `  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )  x.  ( log `  A ) ) )
17217, 170, 171syl2anc 661 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  =  ( (
# `  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )  x.  ( log `  A ) ) )
173 hashcl 12129 . . . . . . 7  |-  ( ( ( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin  ->  ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  e. 
NN0 )
17417, 173syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  NN0 )
175174nn0red 10640 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  RR )
176 logge0 22057 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( log `  A ) )
177 reflcl 11649 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  e.  RR )
17815, 177syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  RR )
179 fzfid 11798 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 1 ... ( |_ `  ( sqr `  A
) ) )  e. 
Fin )
180 ppisval 22444 . . . . . . . . . . 11  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A
) ) )  i^i 
Prime ) )
18115, 180syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime ) )
182 inss1 3573 . . . . . . . . . . 11  |-  ( ( 2 ... ( |_
`  ( sqr `  A
) ) )  i^i 
Prime )  C_  ( 2 ... ( |_ `  ( sqr `  A ) ) )
183 2nn 10482 . . . . . . . . . . . . 13  |-  2  e.  NN
184 nnuz 10899 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  1 )
185183, 184eleqtri 2515 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  1 )
186 fzss1 11500 . . . . . . . . . . . 12  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  ( sqr `  A ) ) )  C_  (
1 ... ( |_ `  ( sqr `  A ) ) ) )
187185, 186mp1i 12 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 2 ... ( |_ `  ( sqr `  A
) ) )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
188182, 187syl5ss 3370 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
189181, 188eqsstrd 3393 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )
190 ssdomg 7358 . . . . . . . . 9  |-  ( ( 1 ... ( |_
`  ( sqr `  A
) ) )  e. 
Fin  ->  ( ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A ) ) )  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
191179, 189, 190sylc 60 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
192 hashdom 12145 . . . . . . . . 9  |-  ( ( ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  e.  Fin  /\  (
1 ... ( |_ `  ( sqr `  A ) ) )  e.  Fin )  ->  ( ( # `  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) )  <_  ( # `  (
1 ... ( |_ `  ( sqr `  A ) ) ) )  <->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
19317, 179, 192syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  <_ 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  <->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
194191, 193mpbird 232 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( # `
 ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
195 flge0nn0 11669 . . . . . . . . 9  |-  ( ( ( sqr `  A
)  e.  RR  /\  0  <_  ( sqr `  A
) )  ->  ( |_ `  ( sqr `  A
) )  e.  NN0 )
19615, 56, 195syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  NN0 )
197 hashfz1 12120 . . . . . . . 8  |-  ( ( |_ `  ( sqr `  A ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
198196, 197syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
199194, 198breqtrd 4319 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( |_ `  ( sqr `  A
) ) )
200 flle 11652 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  <_  ( sqr `  A ) )
20115, 200syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  <_  ( sqr `  A
) )
202175, 178, 15, 199, 201letrd 9531 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( sqr `  A ) )
203175, 15, 21, 176, 202lemul1ad 10275 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  x.  ( log `  A
) )  <_  (
( sqr `  A
)  x.  ( log `  A ) ) )
204172, 203eqbrtrd 4315 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  <_  ( ( sqr `  A )  x.  ( log `  A
) ) )
2055, 20, 22, 169, 204letrd 9531 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_ 
( ( sqr `  A
)  x.  ( log `  A ) ) )
2062, 4, 22lesubadd2d 9941 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( (ψ `  A )  -  ( theta `  A ) )  <_  ( ( sqr `  A )  x.  ( log `  A ) )  <-> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) ) )
207205, 206mpbid 210 1  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    \ cdif 3328    i^i cin 3330    C_ wss 3331   class class class wbr 4295   ` cfv 5421  (class class class)co 6094    ~<_ cdom 7311   Fincfn 7313   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    x. cmul 9290    < clt 9421    <_ cle 9422    - cmin 9598    / cdiv 9996   NNcn 10325   2c2 10374   NN0cn0 10582   ZZcz 10649   ZZ>=cuz 10864   RR+crp 10994   [,]cicc 11306   ...cfz 11440   |_cfl 11643   ^cexp 11868   #chash 12106   sqrcsqr 12725   sum_csu 13166   Primecprime 13766   logclog 22009   thetaccht 22431  ψcchp 22433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-fi 7664  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-xneg 11092  df-xadd 11093  df-xmul 11094  df-ioo 11307  df-ioc 11308  df-ico 11309  df-icc 11310  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-fac 12055  df-bc 12082  df-hash 12107  df-shft 12559  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-limsup 12952  df-clim 12969  df-rlim 12970  df-sum 13167  df-ef 13356  df-sin 13358  df-cos 13359  df-pi 13361  df-dvds 13539  df-gcd 13694  df-prm 13767  df-pc 13907  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-hom 14265  df-cco 14266  df-rest 14364  df-topn 14365  df-0g 14383  df-gsum 14384  df-topgen 14385  df-pt 14386  df-prds 14389  df-xrs 14443  df-qtop 14448  df-imas 14449  df-xps 14451  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-submnd 15468  df-mulg 15551  df-cntz 15838  df-cmn 16282  df-psmet 17812  df-xmet 17813  df-met 17814  df-bl 17815  df-mopn 17816  df-fbas 17817  df-fg 17818  df-cnfld 17822  df-top 18506  df-bases 18508  df-topon 18509  df-topsp 18510  df-cld 18626  df-ntr 18627  df-cls 18628  df-nei 18705  df-lp 18743  df-perf 18744  df-cn 18834  df-cnp 18835  df-haus 18922  df-tx 19138  df-hmeo 19331  df-fil 19422  df-fm 19514  df-flim 19515  df-flf 19516  df-xms 19898  df-ms 19899  df-tms 19900  df-cncf 20457  df-limc 21344  df-dv 21345  df-log 22011  df-cht 22437  df-vma 22438  df-chp 22439
This theorem is referenced by:  chpchtlim  22731
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