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 ) ) |
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 ) ) |