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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 36456 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
8 | 1, 4 | dnicld2 36439 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℝ) |
9 | 8 | recnd 11318 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℂ) |
10 | 1, 2 | dnicld2 36439 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) |
11 | 10 | recnd 11318 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℂ) |
12 | 9, 11 | abssubd 15502 | . . . 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 36456 | . . . 4 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
18 | 2 | recnd 11318 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 4 | recnd 11318 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 18, 19 | abssubd 15502 | . . . . 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 5192 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐵 − 𝐴))) |
23 | 13, 22 | eqbrtrd 5188 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
24 | halfre 12507 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
26 | 2, 25 | readdcld 11319 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
27 | reflcl 13847 | . . . 4 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
28 | 26, 27 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
29 | 4, 25 | readdcld 11319 | . . . 4 ⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
30 | reflcl 13847 | . . . 4 ⊢ ((𝐵 + (1 / 2)) ∈ ℝ → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) |
32 | 28, 31 | letrid 11442 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))) ∨ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2))))) |
33 | 7, 23, 32 | mpjaodan 959 | 1 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 1c1 11185 + caddc 11187 ≤ cle 11325 − cmin 11520 / cdiv 11947 2c2 12348 ⌊cfl 13841 abscabs 15283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 |
This theorem is referenced by: dnicn 36458 knoppndvlem11 36488 |
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