Mathbox for Asger C. Ipsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dnibndlem1 Structured version   Visualization version   GIF version

Theorem dnibndlem1 31638
 Description: Lemma for dnibnd 31651. (Contributed by Asger C. Ipsen, 4-Apr-2021.)
Hypotheses
Ref Expression
dnibndlem1.1 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
dnibndlem1.2 (𝜑𝐴 ∈ ℝ)
dnibndlem1.3 (𝜑𝐵 ∈ ℝ)
Assertion
Ref Expression
dnibndlem1 (𝜑 → ((abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem dnibndlem1
StepHypRef Expression
1 dnibndlem1.3 . . . . 5 (𝜑𝐵 ∈ ℝ)
2 dnibndlem1.1 . . . . . 6 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
32dnival 31631 . . . . 5 (𝐵 ∈ ℝ → (𝑇𝐵) = (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)))
41, 3syl 17 . . . 4 (𝜑 → (𝑇𝐵) = (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)))
5 dnibndlem1.2 . . . . 5 (𝜑𝐴 ∈ ℝ)
62dnival 31631 . . . . 5 (𝐴 ∈ ℝ → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
75, 6syl 17 . . . 4 (𝜑 → (𝑇𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))
84, 7oveq12d 6567 . . 3 (𝜑 → ((𝑇𝐵) − (𝑇𝐴)) = ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))))
98fveq2d 6107 . 2 (𝜑 → (abs‘((𝑇𝐵) − (𝑇𝐴))) = (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))))
109breq1d 4593 1 (𝜑 → ((abs‘((𝑇𝐵) − (𝑇𝐴))) ≤ 𝑆 ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  1c1 9816   + caddc 9818   ≤ cle 9954   − cmin 10145   / cdiv 10563  2c2 10947  ⌊cfl 12453  abscabs 13822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  dnibndlem2  31639  dnibndlem9  31646  dnibndlem12  31649
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