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Theorem chpub 23621
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 23524 . . . . 5  |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
21adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  e.  RR )
3 chtcl 23509 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
43adantr 465 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  e.  RR )
52, 4resubcld 10008 . . 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 9614 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  e.  RR )
8 1red 9628 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  e.  RR )
9 0lt1 10096 . . . . . . . . . 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 9759 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  A )
136, 12elrpd 11279 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR+ )
1413rpge0d 11285 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  A )
156, 14resqrtcld 13261 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  e.  RR )
16 ppifi 23505 . . . . 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 23134 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
2017, 19fsumrecl 13568 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  e.  RR )
2113relogcld 23134 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  RR )
2215, 21remulcld 9641 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  x.  ( log `  A ) )  e.  RR )
23 ppifi 23505 . . . . . . 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 3715 . . . . . . . . . . . 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 3497 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  Prime )
28 prmnn 14232 . . . . . . . . . . 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 11280 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR+ )
3130relogcld 23134 . . . . . . . 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 10571 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR )
34 prmuz2 14247 . . . . . . . . . . . . 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 11179 . . . . . . . . . . . . 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 23140 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR+ )
4032, 39rerpdivcld 11308 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
41 reflcl 11936 . . . . . . . . 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 9641 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
4443recnd 9639 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  CC )
4531recnd 9639 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  CC )
4624, 44, 45fsumsub 13615 . . . . 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 10646 . . . . . . . . 9  |-  0  <_  0
4847a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  0 )
498, 6, 6, 14, 11lemul2ad 10506 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  <_  ( A  x.  A ) )
506recnd 9639 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  CC )
5150sqsqrtd 13282 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
5250mulid1d 9630 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  =  A )
5351, 52eqtr4d 2501 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  ( A  x.  1 ) )
5450sqvald 12310 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
5549, 53, 543brtr4d 4486 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  <_  ( A ^
2 ) )
566, 14sqrtge0d 13264 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( sqr `  A ) )
5715, 6, 56, 14le2sqd 12348 . . . . . . . . 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 11617 . . . . . . . 8  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
0  /\  ( sqr `  A )  <_  A
) )  ->  (
0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
) )
607, 6, 48, 58, 59syl22anc 1229 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 0 [,] ( sqr `  A ) ) 
C_  ( 0 [,] A ) )
61 ssrin 3719 . . . . . . 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 3499 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6443, 31resubcld 10008 . . . . . . . 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 9639 . . . . . . 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 3622 . . . . . . . . . . . . . . 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 9631 . . . . . . . . . . . . 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 3714 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
7170, 26sseldi 3497 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
72 0re 9613 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
736adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  A  e.  RR )
74 elicc2 11614 . . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . 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 23138 . . . . . . . . . . . . . 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 4476 . . . . . . . . . . . 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 9628 . . . . . . . . . . . . 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 11326 . . . . . . . . . . . 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 9639 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  CC )
9190sqsqrtd 13282 . . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . . . . . . . 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 3683 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
( p  e.  ( 0 [,] ( sqr `  A ) )  /\  p  e.  Prime ) )
9695rbaib 906 . . . . . . . . . . . . . . . . . . . . 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 9614 . . . . . . . . . . . . . . . . . . . . 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 10571 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR )
10279rpge0d 11285 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  p )
103 elicc2 11614 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A ) ) ) )
104 df-3an 975 . . . . . . . . . . . . . . . . . . . . . . 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 909 . . . . . . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . . . . . . 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 9749 . . . . . . . . . . . . . . . . . 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 12347 . . . . . . . . . . . . . . . . 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 4478 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  <  ( p ^ 2 ) )
116100nnsqcld 12333 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  NN )
117116nnrpd 11280 . . . . . . . . . . . . . . . 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 23110 . . . . . . . . . . . . . . . 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 10917 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
122 relogexp 23106 . . . . . . . . . . . . . . 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 4480 . . . . . . . . . . . . 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 10626 . . . . . . . . . . . . . . 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 11329 . . . . . . . . . . . . 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 10615 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
130128, 129syl6breq 4495 . . . . . . . . . . 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 10915 . . . . . . . . . . . 12  |-  1  e.  ZZ
133 flbi 11955 . . . . . . . . . . . 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 922 . . . . . . . . . 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 6312 . . . . . . . . 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 9630 . . . . . . . . 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 2498 . . . . . . . 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 6311 . . . . . . 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 9938 . . . . . . 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 2498 . . . . . 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 13559 . . . . 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 23619 . . . . . . 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 23510 . . . . . . 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 6314 . . . . 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 2509 . . . 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 11285 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
0  <_  ( log `  p ) )
153 inss2 3715 . . . . . . . . . . . 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 3497 . . . . . . . . . . 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 11280 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  RR+ )
158157relogcld 23134 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
159150, 158subge02d 10165 . . . . . . 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 11939 . . . . . . . 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 11327 . . . . . . 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 9756 . . . . 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 13625 . . . 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 4476 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
17021recnd 9639 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  CC )
171 fsumconst 13617 . . . . 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 12431 . . . . . . 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 10874 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  RR )
176 logge0 23116 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( log `  A ) )
177 reflcl 11936 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  e.  RR )
17815, 177syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  RR )
179 fzfid 12086 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 1 ... ( |_ `  ( sqr `  A
) ) )  e. 
Fin )
180 ppisval 23503 . . . . . . . . . . 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 3714 . . . . . . . . . . 11  |-  ( ( 2 ... ( |_
`  ( sqr `  A
) ) )  i^i 
Prime )  C_  ( 2 ... ( |_ `  ( sqr `  A ) ) )
183 2eluzge1 11152 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  1 )
184 fzss1 11748 . . . . . . . . . . . 12  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  ( sqr `  A ) ) )  C_  (
1 ... ( |_ `  ( sqr `  A ) ) ) )
185183, 184mp1i 12 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 2 ... ( |_ `  ( sqr `  A
) ) )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
186182, 185syl5ss 3510 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
187181, 186eqsstrd 3533 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )
188 ssdomg 7580 . . . . . . . . 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 60 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
190 hashdom 12450 . . . . . . . . 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 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 ) ) ) ) )
192189, 191mpbird 232 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( # `
 ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
193 flge0nn0 11957 . . . . . . . . 9  |-  ( ( ( sqr `  A
)  e.  RR  /\  0  <_  ( sqr `  A
) )  ->  ( |_ `  ( sqr `  A
) )  e.  NN0 )
19415, 56, 193syl2anc 661 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  NN0 )
195 hashfz1 12422 . . . . . . . 8  |-  ( ( |_ `  ( sqr `  A ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
196194, 195syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
197192, 196breqtrd 4480 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( |_ `  ( sqr `  A
) ) )
198 flle 11939 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  <_  ( sqr `  A ) )
19915, 198syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  <_  ( sqr `  A
) )
200175, 178, 15, 197, 199letrd 9756 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( sqr `  A ) )
201175, 15, 21, 176, 200lemul1ad 10505 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  x.  ( log `  A
) )  <_  (
( sqr `  A
)  x.  ( log `  A ) ) )
202172, 201eqbrtrd 4476 . . 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 9756 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_ 
( ( sqr `  A
)  x.  ( log `  A ) ) )
2042, 4, 22lesubadd2d 10172 . 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 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 973    = wceq 1395    e. wcel 1819    \ cdif 3468    i^i cin 3470    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296    ~<_ cdom 7533   Fincfn 7535   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   RR+crp 11245   [,]cicc 11557   ...cfz 11697   |_cfl 11930   ^cexp 12169   #chash 12408   sqrcsqrt 13078   sum_csu 13520   Primecprime 14229   logclog 23068   thetaccht 23490  ψcchp 23492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-cht 23496  df-vma 23497  df-chp 23498
This theorem is referenced by:  chpchtlim  23790
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