Metamath Proof Explorer


Theorem chpval

Description: Value of the second Chebyshev function, or summary von Mangoldt function. (Contributed by Mario Carneiro, 7-Apr-2016)

Ref Expression
Assertion chpval ( 𝐴 ∈ ℝ → ( ψ ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) )

Proof

Step Hyp Ref Expression
1 fveq2 ( 𝑥 = 𝐴 → ( ⌊ ‘ 𝑥 ) = ( ⌊ ‘ 𝐴 ) )
2 1 oveq2d ( 𝑥 = 𝐴 → ( 1 ... ( ⌊ ‘ 𝑥 ) ) = ( 1 ... ( ⌊ ‘ 𝐴 ) ) )
3 2 sumeq1d ( 𝑥 = 𝐴 → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( Λ ‘ 𝑛 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) )
4 df-chp ψ = ( 𝑥 ∈ ℝ ↦ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( Λ ‘ 𝑛 ) )
5 sumex Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) ∈ V
6 3 4 5 fvmpt ( 𝐴 ∈ ℝ → ( ψ ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑛 ) )