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Theorem pntsval 21134
Description: Define the "Selberg function", whose asymptotic behavior is the content of selberg 21110. (Contributed by Mario Carneiro, 31-May-2016.)
Hypothesis
Ref Expression
pntsval.1  |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) ) )
Assertion
Ref Expression
pntsval  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
Distinct variable groups:    i, a, n, A    S, n
Allowed substitution hints:    S( i, a)

Proof of Theorem pntsval
StepHypRef Expression
1 fveq2 5669 . . . . 5  |-  ( i  =  n  ->  (Λ `  i )  =  (Λ `  n ) )
2 fveq2 5669 . . . . . 6  |-  ( i  =  n  ->  ( log `  i )  =  ( log `  n
) )
3 oveq2 6029 . . . . . . 7  |-  ( i  =  n  ->  (
a  /  i )  =  ( a  /  n ) )
43fveq2d 5673 . . . . . 6  |-  ( i  =  n  ->  (ψ `  ( a  /  i
) )  =  (ψ `  ( a  /  n
) ) )
52, 4oveq12d 6039 . . . . 5  |-  ( i  =  n  ->  (
( log `  i
)  +  (ψ `  ( a  /  i
) ) )  =  ( ( log `  n
)  +  (ψ `  ( a  /  n
) ) ) )
61, 5oveq12d 6039 . . . 4  |-  ( i  =  n  ->  (
(Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) )  =  ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
a  /  n ) ) ) ) )
76cbvsumv 12418 . . 3  |-  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
)  x.  ( ( log `  i )  +  (ψ `  (
a  /  i ) ) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  a ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( a  /  n
) ) ) )
8 fveq2 5669 . . . . 5  |-  ( a  =  A  ->  ( |_ `  a )  =  ( |_ `  A
) )
98oveq2d 6037 . . . 4  |-  ( a  =  A  ->  (
1 ... ( |_ `  a ) )  =  ( 1 ... ( |_ `  A ) ) )
10 oveq1 6028 . . . . . . . 8  |-  ( a  =  A  ->  (
a  /  n )  =  ( A  /  n ) )
1110fveq2d 5673 . . . . . . 7  |-  ( a  =  A  ->  (ψ `  ( a  /  n
) )  =  (ψ `  ( A  /  n
) ) )
1211oveq2d 6037 . . . . . 6  |-  ( a  =  A  ->  (
( log `  n
)  +  (ψ `  ( a  /  n
) ) )  =  ( ( log `  n
)  +  (ψ `  ( A  /  n
) ) ) )
1312oveq2d 6037 . . . . 5  |-  ( a  =  A  ->  (
(Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( a  /  n
) ) ) )  =  ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
1413adantr 452 . . . 4  |-  ( ( a  =  A  /\  n  e.  ( 1 ... ( |_ `  a ) ) )  ->  ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
a  /  n ) ) ) )  =  ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
159, 14sumeq12dv 12428 . . 3  |-  ( a  =  A  ->  sum_ n  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
a  /  n ) ) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  A ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( A  /  n
) ) ) ) )
167, 15syl5eq 2432 . 2  |-  ( a  =  A  ->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i
)  x.  ( ( log `  i )  +  (ψ `  (
a  /  i ) ) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  A ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( A  /  n
) ) ) ) )
17 pntsval.1 . 2  |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) ) )
18 sumex 12409 . 2  |-  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) )  e.  _V
1916, 17, 18fvmpt 5746 1  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    e. cmpt 4208   ` cfv 5395  (class class class)co 6021   RRcr 8923   1c1 8925    + caddc 8927    x. cmul 8929    / cdiv 9610   ...cfz 10976   |_cfl 11129   sum_csu 12407   logclog 20320  Λcvma 20742  ψcchp 20743
This theorem is referenced by:  selbergs  21136  selbergsb  21137  pntsval2  21138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-sum 12408
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