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Theorem chpub 20996
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 20899 . . . . 5  |-  ( A  e.  RR  ->  (ψ `  A )  e.  RR )
21adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  e.  RR )
3 chtcl 20884 . . . . 5  |-  ( A  e.  RR  ->  ( theta `  A )  e.  RR )
43adantr 452 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  e.  RR )
52, 4resubcld 9457 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  e.  RR )
6 simpl 444 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR )
7 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
87a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  e.  RR )
9 1re 9082 . . . . . . . . . 10  |-  1  e.  RR
109a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  e.  RR )
11 0lt1 9542 . . . . . . . . . 10  |-  0  <  1
1211a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  1 )
13 simpr 448 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
1  <_  A )
148, 10, 6, 12, 13ltletrd 9222 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <  A )
156, 14elrpd 10638 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  RR+ )
1615rpge0d 10644 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  A )
176, 16resqrcld 12212 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  e.  RR )
18 ppifi 20880 . . . . 5  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin )
1917, 18syl 16 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  e.  Fin )
2015adantr 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  A  e.  RR+ )
2120relogcld 20510 . . . 4  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  A
)  e.  RR )
2219, 21fsumrecl 12520 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  e.  RR )
2315relogcld 20510 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  RR )
2417, 23remulcld 9108 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  x.  ( log `  A ) )  e.  RR )
25 ppifi 20880 . . . . . . 7  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
2625adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] A )  i^i  Prime )  e.  Fin )
27 inss2 3554 . . . . . . . . . . . 12  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
28 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A
)  i^i  Prime ) )
2927, 28sseldi 3338 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  Prime )
30 prmnn 13074 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
3129, 30syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  NN )
3231nnrpd 10639 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR+ )
3332relogcld 20510 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR )
3423adantr 452 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  A
)  e.  RR )
3531nnred 10007 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  RR )
36 prmuz2 13089 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
3729, 36syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  (
ZZ>= `  2 ) )
38 eluz2b2 10540 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
3938simprbi 451 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
4037, 39syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  1  <  p
)
4135, 40rplogcld 20516 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  RR+ )
4234, 41rerpdivcld 10667 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
43 reflcl 11197 . . . . . . . . 9  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4442, 43syl 16 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  RR )
4533, 44remulcld 9108 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
4645recnd 9106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  CC )
4733recnd 9106 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( log `  p
)  e.  CC )
4826, 46, 47fsumsub 12563 . . . . 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
) ) )
49 0le0 10073 . . . . . . . . 9  |-  0  <_  0
5049a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  0 )
5110, 6, 6, 16, 13lemul2ad 9943 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  <_  ( A  x.  A ) )
526recnd 9106 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  A  e.  CC )
5352sqsqrd 12233 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  A )
5452mulid1d 9097 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A  x.  1 )  =  A )
5553, 54eqtr4d 2470 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  =  ( A  x.  1 ) )
5652sqvald 11512 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( A ^ 2 )  =  ( A  x.  A ) )
5751, 55, 563brtr4d 4234 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
) ^ 2 )  <_  ( A ^
2 ) )
586, 16sqrge0d 12215 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( sqr `  A ) )
5917, 6, 58, 16le2sqd 11550 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( sqr `  A
)  <_  A  <->  ( ( sqr `  A ) ^
2 )  <_  ( A ^ 2 ) ) )
6057, 59mpbird 224 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( sqr `  A
)  <_  A )
61 iccss 10970 . . . . . . . 8  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
0  /\  ( sqr `  A )  <_  A
) )  ->  (
0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
) )
628, 6, 50, 60, 61syl22anc 1185 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 0 [,] ( sqr `  A ) ) 
C_  ( 0 [,] A ) )
63 ssrin 3558 . . . . . . 7  |-  ( ( 0 [,] ( sqr `  A ) )  C_  ( 0 [,] A
)  ->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  ( ( 0 [,] A )  i^i  Prime ) )
6462, 63syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( ( 0 [,] A )  i^i 
Prime ) )
6564sselda 3340 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
6645, 33resubcld 9457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  RR )
6766recnd 9106 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( ( ( log `  p )  x.  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) )  -  ( log `  p ) )  e.  CC )
6865, 67syldan 457 . . . . . 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 )
69 eldifi 3461 . . . . . . . . . . . . . . 15  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] A )  i^i  Prime ) )
7069, 47sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  CC )
7170mulid2d 9098 . . . . . . . . . . . . 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
) )
72 inss1 3553 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
7372, 28sseldi 3338 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  e.  ( 0 [,] A ) )
746adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  A  e.  RR )
75 elicc2 10967 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
767, 74, 75sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( p  e.  ( 0 [,] A
)  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) ) )
7773, 76mpbid 202 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) )
7877simp3d 971 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] A
)  i^i  Prime ) )  ->  p  <_  A
)
7969, 78sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  <_  A )
8069, 32sylan2 461 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR+ )
8115adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR+ )
8280, 81logled 20514 . . . . . . . . . . . . . 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 ) ) )
8379, 82mpbid 202 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  <_ 
( log `  A
) )
8471, 83eqbrtrd 4224 . . . . . . . . . . . 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
) )
859a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  e.  RR )
8623adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  A )  e.  RR )
8769, 41sylan2 461 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  p )  e.  RR+ )
8885, 86, 87lemuldivd 10685 . . . . . . . . . . . 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 ) ) ) )
8984, 88mpbid 202 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  1  <_  ( ( log `  A
)  /  ( log `  p ) ) )
906adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  RR )
9190recnd 9106 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  e.  CC )
9291sqsqrd 12233 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
) ^ 2 )  =  A )
93 eldifn 3462 . . . . . . . . . . . . . . . . . . . 20  |-  ( p  e.  ( ( ( 0 [,] A )  i^i  Prime )  \  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  -.  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )
9493adantl 453 . . . . . . . . . . . . . . . . . . 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 ) )
9569, 29sylan2 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  Prime )
96 elin 3522 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
( p  e.  ( 0 [,] ( sqr `  A ) )  /\  p  e.  Prime ) )
9796rbaib 874 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  <-> 
p  e.  ( 0 [,] ( sqr `  A
) ) ) )
9895, 97syl 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 ) ) ) )
997a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  e.  RR )
10017adantr 452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( sqr `  A )  e.  RR )
10169, 31sylan2 461 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  NN )
102101nnred 10007 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  p  e.  RR )
10380rpge0d 10644 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  p )
104 elicc2 10967 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A ) ) ) )
105 df-3an 938 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  ( sqr `  A
) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) )
106104, 105syl6bb 253 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( 0  e.  RR  /\  ( sqr `  A )  e.  RR )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  ( sqr `  A
) ) ) )
107106baibd 876 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( 0  e.  RR  /\  ( sqr `  A
)  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] ( sqr `  A ) )  <->  p  <_  ( sqr `  A ) ) )
10899, 100, 102, 103, 107syl22anc 1185 . . . . . . . . . . . . . . . . . . . 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 ) ) )
10998, 108bitrd 245 . . . . . . . . . . . . . . . . . . 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 ) ) )
11094, 109mtbid 292 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  -.  p  <_  ( sqr `  A
) )
111100, 102ltnled 9212 . . . . . . . . . . . . . . . . . 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
) ) )
112110, 111mpbird 224 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( sqr `  A )  < 
p )
11358adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  0  <_  ( sqr `  A
) )
114100, 102, 113, 103lt2sqd 11549 . . . . . . . . . . . . . . . . 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 ) ) )
115112, 114mpbid 202 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( sqr `  A
) ^ 2 )  <  ( p ^
2 ) )
11692, 115eqbrtrrd 4226 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  A  <  ( p ^ 2 ) )
117101nnsqcld 11535 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  NN )
118117nnrpd 10639 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
p ^ 2 )  e.  RR+ )
119 logltb 20486 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR+  /\  (
p ^ 2 )  e.  RR+ )  ->  ( A  <  ( p ^
2 )  <->  ( log `  A )  <  ( log `  ( p ^
2 ) ) ) )
12081, 118, 119syl2anc 643 . . . . . . . . . . . . . . 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 ) ) ) )
121116, 120mpbid 202 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( log `  A )  < 
( log `  (
p ^ 2 ) ) )
122 2z 10304 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
123 relogexp 20482 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  RR+  /\  2  e.  ZZ )  ->  ( log `  ( p ^
2 ) )  =  ( 2  x.  ( log `  p ) ) )
12480, 122, 123sylancl 644 . . . . . . . . . . . . . 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 ) ) )
125121, 124breqtrd 4228 . . . . . . . . . . . . 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 ) ) )
126 2re 10061 . . . . . . . . . . . . . . 15  |-  2  e.  RR
127126a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  2  e.  RR )
12886, 127, 87ltdivmul2d 10688 . . . . . . . . . . . . 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
) ) ) )
129125, 128mpbird 224 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  A
)  /  ( log `  p ) )  <  2 )
130 df-2 10050 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
131129, 130syl6breq 4243 . . . . . . . . . . 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 ) )
13269, 42sylan2 461 . . . . . . . . . . . 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 )
133 1z 10303 . . . . . . . . . . . 12  |-  1  e.  ZZ
134 flbi 11215 . . . . . . . . . . . 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 ) ) ) )
135132, 133, 134sylancl 644 . . . . . . . . . . 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 ) ) ) )
13689, 131, 135mpbir2and 889 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  =  1 )
137136oveq2d 6089 . . . . . . . . 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 ) )
13870mulid1d 9097 . . . . . . . . 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
) )
139137, 138eqtrd 2467 . . . . . . . 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 ) )
140139oveq1d 6088 . . . . . . 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 ) ) )
14170subidd 9391 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( ( 0 [,] A )  i^i  Prime ) 
\  ( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) ) )  ->  (
( log `  p
)  -  ( log `  p ) )  =  0 )
142140, 141eqtrd 2467 . . . . . 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 )
14364, 68, 142, 26fsumss 12511 . . . . 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 ) ) )
144 chpval2 20994 . . . . . . 7  |-  ( A  e.  RR  ->  (ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( ( log `  p )  x.  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )
145144adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )
146 chtval 20885 . . . . . . 7  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
147146adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
148145, 147oveq12d 6091 . . . . 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
) ) )
14948, 143, 1483eqtr4rd 2478 . . . 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 ) ) )
15065, 66syldan 457 . . . . 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 )
15165, 45syldan 457 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  e.  RR )
15265, 41syldan 457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR+ )
153152rpge0d 10644 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
0  <_  ( log `  p ) )
154 inss2 3554 . . . . . . . . . . . 12  |-  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime )  C_  Prime
155 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime ) )
156154, 155sseldi 3338 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  Prime )
157156, 30syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  NN )
158157nnrpd 10639 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  ->  p  e.  RR+ )
159158relogcld 20510 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( log `  p
)  e.  RR )
160151, 159subge02d 9610 . . . . . . 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 ) ) ) ) ) )
161153, 160mpbid 202 . . . . . 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 ) ) ) ) )
16265, 42syldan 457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  A
)  /  ( log `  p ) )  e.  RR )
163 flle 11200 . . . . . . . 8  |-  ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
164162, 163syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  <_  ( ( log `  A )  /  ( log `  p ) ) )
16565, 44syldan 457 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( |_ `  (
( log `  A
)  /  ( log `  p ) ) )  e.  RR )
166165, 21, 152lemuldiv2d 10686 . . . . . . 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 ) ) ) )
167164, 166mpbird 224 . . . . . 6  |-  ( ( ( A  e.  RR  /\  1  <_  A )  /\  p  e.  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  -> 
( ( log `  p
)  x.  ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) )  <_  ( log `  A ) )
168150, 151, 21, 161, 167letrd 9219 . . . . 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
) )
16919, 150, 21, 168fsumle 12570 . . . 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
) )
170149, 169eqbrtrd 4224 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
) )
17123recnd 9106 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( log `  A
)  e.  CC )
172 fsumconst 12565 . . . . 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 ) ) )
17319, 171, 172syl2anc 643 . . . 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 ) ) )
174 hashcl 11631 . . . . . . 7  |-  ( ( ( 0 [,] ( sqr `  A ) )  i^i  Prime )  e.  Fin  ->  ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  e. 
NN0 )
17519, 174syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  NN0 )
176175nn0red 10267 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  e.  RR )
177 logge0 20492 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
0  <_  ( log `  A ) )
178 reflcl 11197 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  e.  RR )
17917, 178syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  RR )
180 fzfid 11304 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 1 ... ( |_ `  ( sqr `  A
) ) )  e. 
Fin )
181 ppisval 20878 . . . . . . . . . . 11  |-  ( ( sqr `  A )  e.  RR  ->  (
( 0 [,] ( sqr `  A ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A
) ) )  i^i 
Prime ) )
18217, 181syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime ) )
183 inss1 3553 . . . . . . . . . . 11  |-  ( ( 2 ... ( |_
`  ( sqr `  A
) ) )  i^i 
Prime )  C_  ( 2 ... ( |_ `  ( sqr `  A ) ) )
184 2nn 10125 . . . . . . . . . . . . 13  |-  2  e.  NN
185 nnuz 10513 . . . . . . . . . . . . 13  |-  NN  =  ( ZZ>= `  1 )
186184, 185eleqtri 2507 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  1 )
187 fzss1 11083 . . . . . . . . . . . 12  |-  ( 2  e.  ( ZZ>= `  1
)  ->  ( 2 ... ( |_ `  ( sqr `  A ) ) )  C_  (
1 ... ( |_ `  ( sqr `  A ) ) ) )
188186, 187mp1i 12 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( 2 ... ( |_ `  ( sqr `  A
) ) )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
189183, 188syl5ss 3351 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 2 ... ( |_ `  ( sqr `  A ) ) )  i^i  Prime )  C_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
190182, 189eqsstrd 3374 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime ) 
C_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )
191 ssdomg 7145 . . . . . . . . 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 ) ) ) ) )
192180, 190, 191sylc 58 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( 0 [,] ( sqr `  A
) )  i^i  Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A
) ) ) )
193 hashdom 11645 . . . . . . . . 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 ) ) ) ) )
19419, 180, 193syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  <_ 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  <->  ( (
0 [,] ( sqr `  A ) )  i^i 
Prime )  ~<_  ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
195192, 194mpbird 224 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( # `
 ( 1 ... ( |_ `  ( sqr `  A ) ) ) ) )
196 flge0nn0 11217 . . . . . . . . 9  |-  ( ( ( sqr `  A
)  e.  RR  /\  0  <_  ( sqr `  A
) )  ->  ( |_ `  ( sqr `  A
) )  e.  NN0 )
19717, 58, 196syl2anc 643 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  e.  NN0 )
198 hashfz1 11622 . . . . . . . 8  |-  ( ( |_ `  ( sqr `  A ) )  e. 
NN0  ->  ( # `  (
1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
199197, 198syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( 1 ... ( |_ `  ( sqr `  A ) ) ) )  =  ( |_ `  ( sqr `  A ) ) )
200195, 199breqtrd 4228 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( |_ `  ( sqr `  A
) ) )
201 flle 11200 . . . . . . 7  |-  ( ( sqr `  A )  e.  RR  ->  ( |_ `  ( sqr `  A
) )  <_  ( sqr `  A ) )
20217, 201syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( |_ `  ( sqr `  A ) )  <_  ( sqr `  A
) )
203176, 179, 17, 200, 202letrd 9219 . . . . 5  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( # `  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) )  <_  ( sqr `  A ) )
204176, 17, 23, 177, 203lemul1ad 9942 . . . 4  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( # `  (
( 0 [,] ( sqr `  A ) )  i^i  Prime ) )  x.  ( log `  A
) )  <_  (
( sqr `  A
)  x.  ( log `  A ) ) )
205173, 204eqbrtrd 4224 . . 3  |-  ( ( A  e.  RR  /\  1  <_  A )  ->  sum_ p  e.  ( ( 0 [,] ( sqr `  A ) )  i^i 
Prime ) ( log `  A
)  <_  ( ( sqr `  A )  x.  ( log `  A
) ) )
2065, 22, 24, 170, 205letrd 9219 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( (ψ `  A
)  -  ( theta `  A ) )  <_ 
( ( sqr `  A
)  x.  ( log `  A ) ) )
2072, 4, 24lesubadd2d 9617 . 2  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
( ( (ψ `  A )  -  ( theta `  A ) )  <_  ( ( sqr `  A )  x.  ( log `  A ) )  <-> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) ) )
208206, 207mpbid 202 1  |-  ( ( A  e.  RR  /\  1  <_  A )  -> 
(ψ `  A )  <_  ( ( theta `  A
)  +  ( ( sqr `  A )  x.  ( log `  A
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309    i^i cin 3311    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   RR+crp 10604   [,]cicc 10911   ...cfz 11035   |_cfl 11193   ^cexp 11374   #chash 11610   sqrcsqr 12030   sum_csu 12471   Primecprime 13071   logclog 20444   thetaccht 20865  ψcchp 20867
This theorem is referenced by:  chpchtlim  21165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cht 20871  df-vma 20872  df-chp 20873
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