Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnival | Structured version Visualization version GIF version |
Description: Value of the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnival.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
Ref | Expression |
---|---|
dnival | ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 7214 | . . . 4 ⊢ (𝑥 = 𝐴 → (⌊‘(𝑥 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2)))) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | oveq12d 7209 | . . 3 ⊢ (𝑥 = 𝐴 → ((⌊‘(𝑥 + (1 / 2))) − 𝑥) = ((⌊‘(𝐴 + (1 / 2))) − 𝐴)) |
4 | 3 | fveq2d 6699 | . 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 6708 | . 2 ⊢ (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ V | |
7 | 4, 5, 6 | fvmpt 6796 | 1 ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5120 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 1c1 10695 + caddc 10697 − cmin 11027 / cdiv 11454 2c2 11850 ⌊cfl 13330 abscabs 14762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 |
This theorem is referenced by: dnicld2 34339 dnizeq0 34341 dnizphlfeqhlf 34342 dnibndlem1 34344 knoppcnlem4 34362 |
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