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	$$⊢A\in ℝ\to \psi \left(A\right)=\sum _{n=1}^{⌊A⌋}\Lambda \left(n\right)$$

Proof

Step	Hyp	Ref	Expression
1		fveq2	$$⊢x=A\to ⌊x⌋=⌊A⌋$$
2	1	oveq2d	$$⊢x=A\to \left(1\dots ⌊x⌋\right)=\left(1\dots ⌊A⌋\right)$$
3	2	sumeq1d	$$⊢x=A\to \sum _{n=1}^{⌊x⌋}\Lambda \left(n\right)=\sum _{n=1}^{⌊A⌋}\Lambda \left(n\right)$$
4		df-chp	$$⊢\psi =\left(x\in ℝ⟼\sum _{n=1}^{⌊x⌋}\Lambda \left(n\right)\right)$$
5		sumex	$$⊢\sum _{n=1}^{⌊A⌋}\Lambda \left(n\right)\in \mathrm{V}$$
6	3 4 5	fvmpt	$$⊢A\in ℝ\to \psi \left(A\right)=\sum _{n=1}^{⌊A⌋}\Lambda \left(n\right)$$