Mathbox for Asger C. Ipsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnicn | Structured version Visualization version GIF version |
Description: The "distance to nearest integer" function is continuous. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnicn.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
Ref | Expression |
---|---|
dnicn | ⊢ 𝑇 ∈ (ℝ–cn→ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnicn.1 | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | 1 | dnif 33815 | . 2 ⊢ 𝑇:ℝ⟶ℝ |
3 | simpr 487 | . . . 4 ⊢ ((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+) | |
4 | simplr 767 | . . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → 𝑧 ∈ ℝ) | |
5 | 1, 4 | dnicld2 33814 | . . . . . . . . . 10 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (𝑇‘𝑧) ∈ ℝ) |
6 | simplll 773 | . . . . . . . . . . 11 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → 𝑦 ∈ ℝ) | |
7 | 1, 6 | dnicld2 33814 | . . . . . . . . . 10 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (𝑇‘𝑦) ∈ ℝ) |
8 | 5, 7 | resubcld 11070 | . . . . . . . . 9 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → ((𝑇‘𝑧) − (𝑇‘𝑦)) ∈ ℝ) |
9 | 8 | recnd 10671 | . . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → ((𝑇‘𝑧) − (𝑇‘𝑦)) ∈ ℂ) |
10 | 9 | abscld 14798 | . . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) ∈ ℝ) |
11 | 4, 6 | resubcld 11070 | . . . . . . . . 9 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (𝑧 − 𝑦) ∈ ℝ) |
12 | 11 | recnd 10671 | . . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (𝑧 − 𝑦) ∈ ℂ) |
13 | 12 | abscld 14798 | . . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (abs‘(𝑧 − 𝑦)) ∈ ℝ) |
14 | 3 | ad2antrr 724 | . . . . . . . 8 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → 𝑒 ∈ ℝ+) |
15 | 14 | rpred 12434 | . . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → 𝑒 ∈ ℝ) |
16 | 1, 6, 4 | dnibnd 33832 | . . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) ≤ (abs‘(𝑧 − 𝑦))) |
17 | simpr 487 | . . . . . . 7 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (abs‘(𝑧 − 𝑦)) < 𝑒) | |
18 | 10, 13, 15, 16, 17 | lelttrd 10800 | . . . . . 6 ⊢ ((((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) ∧ (abs‘(𝑧 − 𝑦)) < 𝑒) → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒) |
19 | 18 | ex 415 | . . . . 5 ⊢ (((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) ∧ 𝑧 ∈ ℝ) → ((abs‘(𝑧 − 𝑦)) < 𝑒 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)) |
20 | 19 | ralrimiva 3184 | . . . 4 ⊢ ((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) → ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑒 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)) |
21 | breq2 5072 | . . . . 5 ⊢ (𝑑 = 𝑒 → ((abs‘(𝑧 − 𝑦)) < 𝑑 ↔ (abs‘(𝑧 − 𝑦)) < 𝑒)) | |
22 | 21 | rspceaimv 3630 | . . . 4 ⊢ ((𝑒 ∈ ℝ+ ∧ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑒 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)) → ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑑 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)) |
23 | 3, 20, 22 | syl2anc 586 | . . 3 ⊢ ((𝑦 ∈ ℝ ∧ 𝑒 ∈ ℝ+) → ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑑 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)) |
24 | 23 | rgen2 3205 | . 2 ⊢ ∀𝑦 ∈ ℝ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑑 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒) |
25 | ax-resscn 10596 | . . 3 ⊢ ℝ ⊆ ℂ | |
26 | elcncf2 23500 | . . 3 ⊢ ((ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑇 ∈ (ℝ–cn→ℝ) ↔ (𝑇:ℝ⟶ℝ ∧ ∀𝑦 ∈ ℝ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑑 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒)))) | |
27 | 25, 25, 26 | mp2an 690 | . 2 ⊢ (𝑇 ∈ (ℝ–cn→ℝ) ↔ (𝑇:ℝ⟶ℝ ∧ ∀𝑦 ∈ ℝ ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ ℝ ((abs‘(𝑧 − 𝑦)) < 𝑑 → (abs‘((𝑇‘𝑧) − (𝑇‘𝑦))) < 𝑒))) |
28 | 2, 24, 27 | mpbir2an 709 | 1 ⊢ 𝑇 ∈ (ℝ–cn→ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 1c1 10540 + caddc 10542 < clt 10677 − cmin 10872 / cdiv 11299 2c2 11695 ℝ+crp 12392 ⌊cfl 13163 abscabs 14595 –cn→ccncf 23486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-cncf 23488 |
This theorem is referenced by: knoppcnlem10 33843 |
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