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Theorem chpub 22444
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 22347 . . . . 5  |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
21adantr 462 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  e.  RR )
3 chtcl 22332 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
43adantr 462 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  e.  RR )
52, 4resubcld 9764 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  e.  RR )
6 simpl 454 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR )
7 0red 9375 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  e.  RR )
8 1red 9389 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  e.  RR )
9 0lt1 9850 . . . . . . . . . 10  |-  0  <  1
109a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  1 )
11 simpr 458 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  <_  A )
127, 8, 6, 10, 11ltletrd 9519 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  A )
136, 12elrpd 11013 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR+ )
1413rpge0d 11019 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  A )
156, 14resqrcld 12888 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  e.  RR )
16 ppifi 22328 . . . . 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 462 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  A  e.  RR+ )
1918relogcld 21957 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
2017, 19fsumrecl 13195 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  e.  RR )
2113relogcld 21957 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  RR )
2215, 21remulcld 9402 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  x.  ( log `  A ) )  e.  RR )
23 ppifi 22328 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
2423adantr 462 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] A )  i^i  Prime )  e.  Fin )
25 inss2 3559 . . . . . . . . . . . 12  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
26 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A
)  i^i  Prime ) )
2725, 26sseldi 3342 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  Prime )
28 prmnn 13749 . . . . . . . . . . 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 11014 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR+ )
3130relogcld 21957 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR )
3221adantr 462 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  A
)  e.  RR )
3329nnred 10325 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR )
34 prmuz2 13764 . . . . . . . . . . . . 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 10915 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
3736simprbi 461 . . . . . . . . . . . 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 21963 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR+ )
4032, 39rerpdivcld 11042 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
41 reflcl 11630 . . . . . . . . 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 9402 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
4443recnd 9400 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  CC )
4531recnd 9400 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  CC )
4624, 44, 45fsumsub 13238 . . . . 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 10399 . . . . . . . . 9  |-  0  <_  0
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  0 )
498, 6, 6, 14, 11lemul2ad 10261 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  <_  ( A  x.  A ) )
506recnd 9400 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  CC )
5150sqsqrd 12909 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
5250mulid1d 9391 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  =  A )
5351, 52eqtr4d 2468 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  ( A  x.  1 ) )
5450sqvald 11989 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
5549, 53, 543brtr4d 4310 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  <_  ( A ^
2 ) )
566, 14sqrge0d 12891 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( sqr `  A ) )
5715, 6, 56, 14le2sqd 12027 . . . . . . . . 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 11351 . . . . . . . 8  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
0  /\  ( sqr `  A )  <_  A
) )  ->  (
0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
) )
607, 6, 48, 58, 59syl22anc 1212 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 0 [,] ( sqr `  A ) ) 
C_  ( 0 [,] A ) )
61 ssrin 3563 . . . . . . 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 3344 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6443, 31resubcld 9764 . . . . . . . 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 9400 . . . . . . 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 467 . . . . . 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 3466 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6867, 45sylan2 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  CC )
6968mulid2d 9392 . . . . . . . . . . . . 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 3558 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
7170, 26sseldi 3342 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
72 0re 9374 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
736adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  A  e.  RR )
74 elicc2 11348 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
7572, 73, 74sylancr 656 . . . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  <_  A
)
7867, 77sylan2 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  <_  A )
7967, 30sylan2 471 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR+ )
8013adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR+ )
8179, 80logled 21961 . . . . . . . . . . . . . 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 4300 . . . . . . . . . . . 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 9389 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  e.  RR )
8521adantr 462 . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 11060 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR )
9089recnd 9400 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  CC )
9190sqsqrd 12909 . . . . . . . . . . . . . . . 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 3467 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  -.  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )
9392adantl 463 . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  Prime )
95 elin 3527 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
( p  e.  ( 0 [,] ( sqr `  A ) )  /\  p  e.  Prime ) )
9695rbaib 891 . . . . . . . . . . . . . . . . . . . . 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 9375 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  e.  RR )
9915adantr 462 . . . . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  NN )
101100nnred 10325 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR )
10279rpge0d 11019 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  p )
103 elicc2 11348 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A ) ) ) )
104 df-3an 960 . . . . . . . . . . . . . . . . . . . . . . 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 893 . . . . . . . . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . . . . . . 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 9509 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 12026 . . . . . . . . . . . . . . . . 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 4302 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  <  ( p ^ 2 ) )
116100nnsqcld 12012 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  NN )
117116nnrpd 11014 . . . . . . . . . . . . . . . 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 21933 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR+  /\  (
p ^ 2 )  e.  RR+ )  ->  ( A  <  ( p ^
2 )  <->  ( log `  A )  <  ( log `  ( p ^
2 ) ) ) )
11980, 117, 118syl2anc 654 . . . . . . . . . . . . . . 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 10666 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
122 relogexp 21929 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( p ^
2 ) )  =  ( 2  x.  ( log `  p ) ) )
12379, 121, 122sylancl 655 . . . . . . . . . . . . . 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 4304 . . . . . . . . . . . . 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 10379 . . . . . . . . . . . . . . 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 11063 . . . . . . . . . . . . 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 10368 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
130128, 129syl6breq 4319 . . . . . . . . . . 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 471 . . . . . . . . . . . 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 10664 . . . . . . . . . . . 12  |-  1  e.  ZZ
133 flbi 11648 . . . . . . . . . . . 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 655 . . . . . . . . . . 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 906 . . . . . . . . . 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 6096 . . . . . . . . 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 9391 . . . . . . . . 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 2465 . . . . . . . 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 6095 . . . . . . 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 9695 . . . . . . 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 2465 . . . . . 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 13186 . . . . 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 22442 . . . . . . 7  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
144143adantr 462 . . . . . 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 22333 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
146145adantr 462 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
147144, 146oveq12d 6098 . . . . 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 2476 . . . 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 467 . . . . 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 467 . . . . . 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 467 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
152151rpge0d 11019 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
0  <_  ( log `  p ) )
153 inss2 3559 . . . . . . . . . . . 12  |-  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  Prime
154 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )
155153, 154sseldi 3342 . . . . . . . . . . 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 11014 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  RR+ )
158157relogcld 21957 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
159150, 158subge02d 9919 . . . . . . 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 467 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
162 flle 11633 . . . . . . . 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 467 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  RR )
165164, 19, 151lemuldiv2d 11061 . . . . . . 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 9516 . . . . 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 13245 . . . 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 4300 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
17021recnd 9400 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  CC )
171 fsumconst 13240 . . . . 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 654 . . . 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 12110 . . . . . . 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 10625 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  RR )
176 logge0 21939 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( log `  A ) )
177 reflcl 11630 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  e.  RR )
17815, 177syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  RR )
179 fzfid 11779 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 1 ... ( |_ `  ( sqr `  A
) ) )  e. 
Fin )
180 ppisval 22326 . . . . . . . . . . 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 3558 . . . . . . . . . . 11  |-  ( ( 2 ... ( |_
`  ( sqr `  A
) ) )  i^i 
Prime )  C_  ( 2 ... ( |_ `  ( sqr `  A ) ) )
183 2nn 10467 . . . . . . . . . . . . 13  |-  2  e.  NN
184 nnuz 10884 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  1 )
185183, 184eleqtri 2505 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  1 )
186 fzss1 11484 . . . . . . . . . . . 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 3355 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
189181, 188eqsstrd 3378 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )
190 ssdomg 7343 . . . . . . . . 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 12126 . . . . . . . . 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 654 . . . . . . . 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 11650 . . . . . . . . 9  |-  ( ( ( sqr `  A
)  e.  RR  /\  0  <_  ( sqr `  A
) )  ->  ( |_ `  ( sqr `  A
) )  e.  NN0 )
19615, 56, 195syl2anc 654 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  NN0 )
197 hashfz1 12101 . . . . . . . 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 4304 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( |_ `  ( sqr `  A
) ) )
200 flle 11633 . . . . . . 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 9516 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( sqr `  A ) )
203175, 15, 21, 176, 202lemul1ad 10260 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  x.  ( log `  A
) )  <_  (
( sqr `  A
)  x.  ( log `  A ) ) )
204172, 203eqbrtrd 4300 . . 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 9516 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_ 
( ( sqr `  A
)  x.  ( log `  A ) ) )
2062, 4, 22lesubadd2d 9926 . 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 958    = wceq 1362    e. wcel 1755    \ cdif 3313    i^i cin 3315    C_ wss 3316   class class class wbr 4280   ` cfv 5406  (class class class)co 6080    ~<_ cdom 7296   Fincfn 7298   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   RR+crp 10979   [,]cicc 11291   ...cfz 11424   |_cfl 11624   ^cexp 11849   #chash 12087   sqrcsqr 12706   sum_csu 13147   Primecprime 13746   logclog 21891   thetaccht 22313  ψcchp 22315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cht 22319  df-vma 22320  df-chp 22321
This theorem is referenced by:  chpchtlim  22613
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