Theorem dnival 32461

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

Hypothesis

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

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem dnival

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + (1 / 2)) = (𝐴 + (1 / 2)))
2	1	fveq2d 6195	. . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))
3		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	oveq12d 6668	. . 3 ⊢ (𝑥 = 𝐴 → ((⌊‘(𝑥 + (1 / 2))) − 𝑥) = ((⌊‘(𝐴 + (1 / 2))) − 𝐴))
5	4	fveq2d 6195	. 2 ⊢ (𝑥 = 𝐴 → (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
6		dnival.1	. 2 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
7		fvex 6201	. 2 ⊢ (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ V
8	5, 6, 7	fvmpt 6282	1 ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  1c1 9937   + caddc 9939   − cmin 10266   / cdiv 10684  2c2 11070  ⌊cfl 12591  abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  dnicld2  32463  dnizeq0  32465  dnizphlfeqhlf  32466  dnibndlem1  32468  knoppcnlem4  32486