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Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibnd | Structured version Visualization version GIF version |
Description: The "distance to nearest integer" function is 1-Lipschitz continuous, i.e., is a short map. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibnd.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
dnibnd.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dnibnd.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
dnibnd | ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibnd.1 | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | dnibnd.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐴 ∈ ℝ) |
4 | dnibnd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐵 ∈ ℝ) |
6 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) | |
7 | 1, 3, 5, 6 | dnibndlem13 36472 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
8 | 1, 4 | dnicld2 36455 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℝ) |
9 | 8 | recnd 11286 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℂ) |
10 | 1, 2 | dnicld2 36455 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) |
11 | 10 | recnd 11286 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℂ) |
12 | 9, 11 | abssubd 15488 | . . . 4 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((𝑇‘𝐴) − (𝑇‘𝐵)))) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((𝑇‘𝐴) − (𝑇‘𝐵)))) |
14 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → 𝐵 ∈ ℝ) |
15 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → 𝐴 ∈ ℝ) |
16 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) | |
17 | 1, 14, 15, 16 | dnibndlem13 36472 | . . . 4 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
18 | 2 | recnd 11286 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 4 | recnd 11286 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 18, 19 | abssubd 15488 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
21 | 20 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
22 | 17, 21 | breqtrd 5173 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐵 − 𝐴))) |
23 | 13, 22 | eqbrtrd 5169 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
24 | halfre 12477 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
26 | 2, 25 | readdcld 11287 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
27 | reflcl 13832 | . . . 4 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
28 | 26, 27 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
29 | 4, 25 | readdcld 11287 | . . . 4 ⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
30 | reflcl 13832 | . . . 4 ⊢ ((𝐵 + (1 / 2)) ∈ ℝ → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) |
32 | 28, 31 | letrid 11410 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))) ∨ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2))))) |
33 | 7, 23, 32 | mpjaodan 960 | 1 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 1c1 11153 + caddc 11155 ≤ cle 11293 − cmin 11489 / cdiv 11917 2c2 12318 ⌊cfl 13826 abscabs 15269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fl 13828 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 |
This theorem is referenced by: dnicn 36474 knoppndvlem11 36504 |
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