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 ) )