Theorem dnival

Description: Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021)

	Ref	Expression
Hypothesis	dnival.1	⊢ 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) )
Assertion	dnival	⊢ ( 𝐴 ∈ ℝ → ( 𝑇 ‘ 𝐴 ) = ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dnival.1	⊢ 𝑇 = ( 𝑥 ∈ ℝ ↦ ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) )
2		fvoveq1	⊢ ( 𝑥 = 𝐴 → ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) = ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) )
3		id	⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 )
4	2 3	oveq12d	⊢ ( 𝑥 = 𝐴 → ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) = ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) )
5	4	fveq2d	⊢ ( 𝑥 = 𝐴 → ( abs ‘ ( ( ⌊ ‘ ( 𝑥 + ( 1 / 2 ) ) ) − 𝑥 ) ) = ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) )
6		fvex	⊢ ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) ∈ V
7	5 1 6	fvmpt	⊢ ( 𝐴 ∈ ℝ → ( 𝑇 ‘ 𝐴 ) = ( abs ‘ ( ( ⌊ ‘ ( 𝐴 + ( 1 / 2 ) ) ) − 𝐴 ) ) )