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Theorem dnival 33924
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 7162 . . . 4 (𝑥 = 𝐴 → (⌊‘(𝑥 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2))))
2 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
31, 2oveq12d 7157 . . 3 (𝑥 = 𝐴 → ((⌊‘(𝑥 + (1 / 2))) − 𝑥) = ((⌊‘(𝐴 + (1 / 2))) − 𝐴))
43fveq2d 6653 . 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 6662 . 2 (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ V
74, 5, 6fvmpt 6749 1 (𝐴 ∈ ℝ → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cmpt 5113  cfv 6328  (class class class)co 7139  cr 10529  1c1 10531   + caddc 10533  cmin 10863   / cdiv 11290  2c2 11684  cfl 13159  abscabs 14589
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142
This theorem is referenced by:  dnicld2  33926  dnizeq0  33928  dnizphlfeqhlf  33929  dnibndlem1  33931  knoppcnlem4  33949
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