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Theorem dnival 31476
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 oveq1 6432 . . . . 5 (𝑥 = 𝐴 → (𝑥 + (1 / 2)) = (𝐴 + (1 / 2)))
21fveq2d 5990 . . . 4 (𝑥 = 𝐴 → (⌊‘(𝑥 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))
3 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
42, 3oveq12d 6443 . . 3 (𝑥 = 𝐴 → ((⌊‘(𝑥 + (1 / 2))) − 𝑥) = ((⌊‘(𝐴 + (1 / 2))) − 𝐴))
54fveq2d 5990 . 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 5996 . 2 (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ V
85, 6, 7fvmpt 6074 1 (𝐴 ∈ ℝ → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  cmpt 4541  cfv 5689  (class class class)co 6425  cr 9688  1c1 9690   + caddc 9692  cmin 10015   / cdiv 10431  2c2 10823  cfl 12317  abscabs 13676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6428
This theorem is referenced by:  dnicld2  31478  dnizeq0  31480  dnizphlfeqhlf  31481  dnibndlem1  31483  knoppcnlem4  31501
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