Metamath Proof Explorer


Theorem chtval

Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion chtval ( 𝐴 ∈ ℝ → ( θ ‘ 𝐴 ) = Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑝 ) )

Proof

Step Hyp Ref Expression
1 oveq2 ( 𝑥 = 𝐴 → ( 0 [,] 𝑥 ) = ( 0 [,] 𝐴 ) )
2 1 ineq1d ( 𝑥 = 𝐴 → ( ( 0 [,] 𝑥 ) ∩ ℙ ) = ( ( 0 [,] 𝐴 ) ∩ ℙ ) )
3 2 sumeq1d ( 𝑥 = 𝐴 → Σ 𝑝 ∈ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ( log ‘ 𝑝 ) = Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑝 ) )
4 df-cht θ = ( 𝑥 ∈ ℝ ↦ Σ 𝑝 ∈ ( ( 0 [,] 𝑥 ) ∩ ℙ ) ( log ‘ 𝑝 ) )
5 sumex Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑝 ) ∈ V
6 3 4 5 fvmpt ( 𝐴 ∈ ℝ → ( θ ‘ 𝐴 ) = Σ 𝑝 ∈ ( ( 0 [,] 𝐴 ) ∩ ℙ ) ( log ‘ 𝑝 ) )