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 ) )