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 484 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐴 ∈ ℝ) |
4 | dnibnd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐵 ∈ ℝ) |
6 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) | |
7 | 1, 3, 5, 6 | dnibndlem13 33942 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
8 | 1, 4 | dnicld2 33925 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℝ) |
9 | 8 | recnd 10658 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐵) ∈ ℂ) |
10 | 1, 2 | dnicld2 33925 | . . . . . 6 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℝ) |
11 | 10 | recnd 10658 | . . . . 5 ⊢ (𝜑 → (𝑇‘𝐴) ∈ ℂ) |
12 | 9, 11 | abssubd 14805 | . . . 4 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((𝑇‘𝐴) − (𝑇‘𝐵)))) |
13 | 12 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) = (abs‘((𝑇‘𝐴) − (𝑇‘𝐵)))) |
14 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → 𝐵 ∈ ℝ) |
15 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → 𝐴 ∈ ℝ) |
16 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) | |
17 | 1, 14, 15, 16 | dnibndlem13 33942 | . . . 4 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))) |
18 | 2 | recnd 10658 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | 4 | recnd 10658 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 18, 19 | abssubd 14805 | . . . . 5 ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
21 | 20 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴))) |
22 | 17, 21 | breqtrd 5056 | . . 3 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐴) − (𝑇‘𝐵))) ≤ (abs‘(𝐵 − 𝐴))) |
23 | 13, 22 | eqbrtrd 5052 | . 2 ⊢ ((𝜑 ∧ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
24 | halfre 11839 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
26 | 2, 25 | readdcld 10659 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
27 | reflcl 13161 | . . . 4 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
28 | 26, 27 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
29 | 4, 25 | readdcld 10659 | . . . 4 ⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
30 | reflcl 13161 | . . . 4 ⊢ ((𝐵 + (1 / 2)) ∈ ℝ → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) |
32 | 28, 31 | letrid 10781 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) ≤ (⌊‘(𝐵 + (1 / 2))) ∨ (⌊‘(𝐵 + (1 / 2))) ≤ (⌊‘(𝐴 + (1 / 2))))) |
33 | 7, 23, 32 | mpjaodan 956 | 1 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 1c1 10527 + caddc 10529 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ⌊cfl 13155 abscabs 14585 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: dnicn 33944 knoppndvlem11 33974 |
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