Users' Mathboxes Mathbox for Asger C. Ipsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnival Structured version   Visualization version   GIF version

Theorem dnival 34865
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 → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem dnival
StepHypRef Expression
1 fvoveq1 7374 . . . 4 (𝑥 = 𝐴 → (⌊‘(𝑥 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))
2 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2oveq12d 7369 . . 3 (𝑥 = 𝐴 → ((⌊‘(𝑥 + (1 / 2))) − 𝑥) = ((⌊‘(𝐴 + (1 / 2))) − 𝐴))
43fveq2d 6843 . 2 (𝑥 = 𝐴 → (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
5 dnival.1 . 2 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
6 fvex 6852 . 2 (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ V
74, 5, 6fvmpt 6945 1 (𝐴 ∈ ℝ → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5186  cfv 6493  (class class class)co 7351  cr 11008  1c1 11010   + caddc 11012  cmin 11343   / cdiv 11770  2c2 12166  cfl 13649  abscabs 15078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354
This theorem is referenced by:  dnicld2  34867  dnizeq0  34869  dnizphlfeqhlf  34870  dnibndlem1  34872  knoppcnlem4  34890
  Copyright terms: Public domain W3C validator