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Metamath Proof Explorer

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 ) )