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Theorem chpub 24227
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 24130 . . . . 5  |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
21adantr 472 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  e.  RR )
3 chtcl 24115 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
43adantr 472 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  e.  RR )
52, 4resubcld 10068 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  e.  RR )
6 simpl 464 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR )
7 0red 9662 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  e.  RR )
8 1red 9676 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  e.  RR )
9 0lt1 10157 . . . . . . . . . 10  |-  0  <  1
109a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  1 )
11 simpr 468 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  <_  A )
127, 8, 6, 10, 11ltletrd 9812 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  A )
136, 12elrpd 11361 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR+ )
1413rpge0d 11368 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  A )
156, 14resqrtcld 13556 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  e.  RR )
16 ppifi 24111 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin )
1715, 16syl 17 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  e.  Fin )
1813adantr 472 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  A  e.  RR+ )
1918relogcld 23651 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
2017, 19fsumrecl 13877 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  e.  RR )
2113relogcld 23651 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  RR )
2215, 21remulcld 9689 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  x.  ( log `  A ) )  e.  RR )
23 ppifi 24111 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
2423adantr 472 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] A )  i^i  Prime )  e.  Fin )
25 inss2 3644 . . . . . . . . . . . 12  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
26 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A
)  i^i  Prime ) )
2725, 26sseldi 3416 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  Prime )
28 prmnn 14704 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
2927, 28syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  NN )
3029nnrpd 11362 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR+ )
3130relogcld 23651 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR )
3221adantr 472 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  A
)  e.  RR )
3329nnred 10646 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR )
34 prmuz2 14721 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
3527, 34syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  (
ZZ>= `  2 ) )
36 eluz2b2 11254 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
3736simprbi 471 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
3835, 37syl 17 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  1  <  p
)
3933, 38rplogcld 23657 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR+ )
4032, 39rerpdivcld 11392 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
41 reflcl 12065 . . . . . . . . 9  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4240, 41syl 17 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4331, 42remulcld 9689 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
4443recnd 9687 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  CC )
4531recnd 9687 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  CC )
4624, 44, 45fsumsub 13926 . . . . 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 10721 . . . . . . . . 9  |-  0  <_  0
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  0 )
498, 6, 6, 14, 11lemul2ad 10569 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  <_  ( A  x.  A ) )
506recnd 9687 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  CC )
5150sqsqrtd 13578 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
5250mulid1d 9678 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  =  A )
5351, 52eqtr4d 2508 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  ( A  x.  1 ) )
5450sqvald 12451 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
5549, 53, 543brtr4d 4426 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  <_  ( A ^
2 ) )
566, 14sqrtge0d 13559 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( sqr `  A ) )
5715, 6, 56, 14le2sqd 12489 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  <_  A  <->  ( ( sqr `  A ) ^
2 )  <_  ( A ^ 2 ) ) )
5855, 57mpbird 240 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  <_  A )
59 iccss 11727 . . . . . . . 8  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
0  /\  ( sqr `  A )  <_  A
) )  ->  (
0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
) )
607, 6, 48, 58, 59syl22anc 1293 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 0 [,] ( sqr `  A ) ) 
C_  ( 0 [,] A ) )
61 ssrin 3648 . . . . . . 7  |-  ( ( 0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
)  ->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  ( ( 0 [,] A )  i^i  Prime ) )
6260, 61syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( ( 0 [,] A )  i^i 
Prime ) )
6362sselda 3418 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6443, 31resubcld 10068 . . . . . . . 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 9687 . . . . . . 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 478 . . . . . 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 3544 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6867, 45sylan2 482 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  CC )
6968mulid2d 9679 . . . . . . . . . . . . 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 3643 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
7170, 26sseldi 3416 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
72 0re 9661 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
736adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  A  e.  RR )
74 elicc2 11724 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
7572, 73, 74sylancr 676 . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) )
7776simp3d 1044 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  <_  A
)
7867, 77sylan2 482 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  <_  A )
7967, 30sylan2 482 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR+ )
8013adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR+ )
8179, 80logled 23655 . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . 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 4416 . . . . . . . . . . . 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 9676 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  e.  RR )
8521adantr 472 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 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 11410 . . . . . . . . . . . 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 215 . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR )
9089recnd 9687 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  CC )
9190sqsqrtd 13578 . . . . . . . . . . . . . . . 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 3545 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  -.  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )
9392adantl 473 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  Prime )
95 elin 3608 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
( p  e.  ( 0 [,] ( sqr `  A ) )  /\  p  e.  Prime ) )
9695rbaib 922 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
p  e.  ( 0 [,] ( sqr `  A
) ) ) )
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 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 9662 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  e.  RR )
9915adantr 472 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  NN )
101100nnred 10646 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR )
10279rpge0d 11368 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  p )
103 elicc2 11724 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A ) ) ) )
104 df-3an 1009 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A
) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) )
105103, 104syl6bb 269 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) ) )
106105baibd 923 . . . . . . . . . . . . . . . . . . . . 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 1293 . . . . . . . . . . . . . . . . . . . 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 261 . . . . . . . . . . . . . . . . . . 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 307 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  -.  p  <_  ( sqr `  A
) )
11099, 101ltnled 9799 . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( sqr `  A )  < 
p )
11256adantr 472 . . . . . . . . . . . . . . . . . 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 12488 . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . 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 4418 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  <  ( p ^ 2 ) )
116100nnsqcld 12474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  NN )
117116nnrpd 11362 . . . . . . . . . . . . . . . 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 23628 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR+  /\  (
p ^ 2 )  e.  RR+ )  ->  ( A  <  ( p ^
2 )  <->  ( log `  A )  <  ( log `  ( p ^
2 ) ) ) )
11980, 117, 118syl2anc 673 . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . 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 10993 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
122 relogexp 23624 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( p ^
2 ) )  =  ( 2  x.  ( log `  p ) ) )
12379, 121, 122sylancl 675 . . . . . . . . . . . . . 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 4420 . . . . . . . . . . . . 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 10701 . . . . . . . . . . . . . . 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 11413 . . . . . . . . . . . . 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 240 . . . . . . . . . . . 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 10690 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
130128, 129syl6breq 4435 . . . . . . . . . . 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 482 . . . . . . . . . . . 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 10991 . . . . . . . . . . . 12  |-  1  e.  ZZ
133 flbi 12084 . . . . . . . . . . . 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 675 . . . . . . . . . . 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 936 . . . . . . . . . 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 6324 . . . . . . . . 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 9678 . . . . . . . . 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 2505 . . . . . . . 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 6323 . . . . . . 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 9993 . . . . . . 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 2505 . . . . . 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 13868 . . . . 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 24225 . . . . . . 7  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
144143adantr 472 . . . . . 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 24116 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
146145adantr 472 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
147144, 146oveq12d 6326 . . . . 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 2516 . . . 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 478 . . . . 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 478 . . . . . 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 478 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
152151rpge0d 11368 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
0  <_  ( log `  p ) )
153 inss2 3644 . . . . . . . . . . . 12  |-  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  Prime
154 simpr 468 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )
155153, 154sseldi 3416 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  Prime )
156155, 28syl 17 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  NN )
157156nnrpd 11362 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  RR+ )
158157relogcld 23651 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
159150, 158subge02d 10226 . . . . . . 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 215 . . . . . 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 478 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
162 flle 12068 . . . . . . . 8  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
163161, 162syl 17 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
16463, 42syldan 478 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  RR )
165164, 19, 151lemuldiv2d 11411 . . . . . . 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 240 . . . . . 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 9809 . . . . 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 13936 . . . 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 4416 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
17021recnd 9687 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  CC )
171 fsumconst 13928 . . . . 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 673 . . . 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 12576 . . . . . . 7  |-  ( ( ( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin  ->  ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  e. 
NN0 )
17417, 173syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  NN0 )
175174nn0red 10950 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  RR )
176 logge0 23633 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( log `  A ) )
177 reflcl 12065 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  e.  RR )
17815, 177syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  RR )
179 fzfid 12224 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 1 ... ( |_ `  ( sqr `  A
) ) )  e. 
Fin )
180 ppisval 24109 . . . . . . . . . . 11  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A
) ) )  i^i 
Prime ) )
18115, 180syl 17 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime ) )
182 inss1 3643 . . . . . . . . . . 11  |-  ( ( 2 ... ( |_
`  ( sqr `  A
) ) )  i^i 
Prime )  C_  ( 2 ... ( |_ `  ( sqr `  A ) ) )
183 2eluzge1 11228 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  1 )
184 fzss1 11863 . . . . . . . . . . . 12  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  ( sqr `  A ) ) )  C_  (
1 ... ( |_ `  ( sqr `  A ) ) ) )
185183, 184mp1i 13 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 2 ... ( |_ `  ( sqr `  A
) ) )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
186182, 185syl5ss 3429 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
187181, 186eqsstrd 3452 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )
188 ssdomg 7633 . . . . . . . . 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 ) ) ) ) )
189179, 187, 188sylc 61 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
190 hashdom 12596 . . . . . . . . 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 ) ) ) ) )
19117, 179, 190syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  <_ 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  <->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
192189, 191mpbird 240 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( # `
 ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
193 flge0nn0 12087 . . . . . . . . 9  |-  ( ( ( sqr `  A
)  e.  RR  /\  0  <_  ( sqr `  A
) )  ->  ( |_ `  ( sqr `  A
) )  e.  NN0 )
19415, 56, 193syl2anc 673 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  NN0 )
195 hashfz1 12567 . . . . . . . 8  |-  ( ( |_ `  ( sqr `  A ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
196194, 195syl 17 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
197192, 196breqtrd 4420 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( |_ `  ( sqr `  A
) ) )
198 flle 12068 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  <_  ( sqr `  A ) )
19915, 198syl 17 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  <_  ( sqr `  A
) )
200175, 178, 15, 197, 199letrd 9809 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( sqr `  A ) )
201175, 15, 21, 176, 200lemul1ad 10568 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  x.  ( log `  A
) )  <_  (
( sqr `  A
)  x.  ( log `  A ) ) )
202172, 201eqbrtrd 4416 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  <_  ( ( sqr `  A )  x.  ( log `  A
) ) )
2035, 20, 22, 169, 202letrd 9809 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_ 
( ( sqr `  A
)  x.  ( log `  A ) ) )
2042, 4, 22lesubadd2d 10233 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( (ψ `  A )  -  ( theta `  A ) )  <_  ( ( sqr `  A )  x.  ( log `  A ) )  <-> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) ) )
205203, 204mpbid 215 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    \ cdif 3387    i^i cin 3389    C_ wss 3390   class class class wbr 4395   ` cfv 5589  (class class class)co 6308    ~<_ cdom 7585   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   [,]cicc 11663   ...cfz 11810   |_cfl 12059   ^cexp 12310   #chash 12553   sqrcsqrt 13373   sum_csu 13829   Primecprime 14701   logclog 23583   thetaccht 24096  ψcchp 24098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cht 24102  df-vma 24103  df-chp 24104
This theorem is referenced by:  chpchtlim  24396
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