Theorem chpfl

Description: The second Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 9-Apr-2016)

	Ref	Expression
Assertion	chpfl	⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = ( ψ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		flidm	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) = ( ⌊ ‘ 𝐴 ) )
2	1	oveq2d	⊢ ( 𝐴 ∈ ℝ → ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) = ( 1 ... ( ⌊ ‘ 𝐴 ) ) )
3	2	sumeq1d	⊢ ( 𝐴 ∈ ℝ → Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑥 ) )
4		reflcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
5		chpval	⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) )
6	4 5	syl	⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ ( ⌊ ‘ 𝐴 ) ) ) ( Λ ‘ 𝑥 ) )
7		chpval	⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ 𝐴 ) = Σ 𝑥 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( Λ ‘ 𝑥 ) )
8	3 6 7	3eqtr4d	⊢ ( 𝐴 ∈ ℝ → ( ψ ‘ ( ⌊ ‘ 𝐴 ) ) = ( ψ ‘ 𝐴 ) )