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Theorem chtval 26475
Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
chtval (𝐴 ∈ ℝ β†’ (ΞΈβ€˜π΄) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
Distinct variable group:   𝐴,𝑝

Proof of Theorem chtval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 oveq2 7366 . . . 4 (π‘₯ = 𝐴 β†’ (0[,]π‘₯) = (0[,]𝐴))
21ineq1d 4172 . . 3 (π‘₯ = 𝐴 β†’ ((0[,]π‘₯) ∩ β„™) = ((0[,]𝐴) ∩ β„™))
32sumeq1d 15591 . 2 (π‘₯ = 𝐴 β†’ Σ𝑝 ∈ ((0[,]π‘₯) ∩ β„™)(logβ€˜π‘) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
4 df-cht 26462 . 2 ΞΈ = (π‘₯ ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]π‘₯) ∩ β„™)(logβ€˜π‘))
5 sumex 15578 . 2 Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘) ∈ V
63, 4, 5fvmpt 6949 1 (𝐴 ∈ ℝ β†’ (ΞΈβ€˜π΄) = Σ𝑝 ∈ ((0[,]𝐴) ∩ β„™)(logβ€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   ∩ cin 3910  β€˜cfv 6497  (class class class)co 7358  β„cr 11055  0cc0 11056  [,]cicc 13273  Ξ£csu 15576  β„™cprime 16552  logclog 25926  ΞΈccht 26456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seq 13913  df-sum 15577  df-cht 26462
This theorem is referenced by:  efchtcl  26476  chtge0  26477  chtfl  26514  chtprm  26518  chtnprm  26519  chtwordi  26521  chtdif  26523  cht1  26530  prmorcht  26543  chtlepsi  26570  chtleppi  26574  chpchtsum  26583  chpub  26584  chtppilimlem1  26837  chtvalz  33299
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