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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem12 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 36097. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem12.1 | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
dnibndlem12.2 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dnibndlem12.3 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dnibndlem12.4 | ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) |
Ref | Expression |
---|---|
dnibndlem12 | ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibndlem12.3 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | dnicld1 36078 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) ∈ ℝ) |
3 | dnibndlem12.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | dnicld1 36078 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
5 | 2, 4 | resubcld 11674 | . . . . 5 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℝ) |
6 | 5 | recnd 11274 | . . . 4 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℂ) |
7 | 6 | abscld 15419 | . . 3 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ∈ ℝ) |
8 | 1red 11247 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1, 3 | resubcld 11674 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 9 | recnd 11274 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
11 | 10 | abscld 15419 | . . 3 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ) |
12 | 8 | rehalfcld 12492 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
13 | 3, 1 | dnibndlem11 36094 | . . . 4 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2)) |
14 | halflt1 12463 | . . . . . 6 ⊢ (1 / 2) < 1 | |
15 | halfre 12459 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
16 | 1re 11246 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
17 | 15, 16 | pm3.2i 469 | . . . . . . 7 ⊢ ((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) |
18 | ltle 11334 | . . . . . . 7 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2) < 1 → (1 / 2) ≤ 1)) | |
19 | 17, 18 | ax-mp 5 | . . . . . 6 ⊢ ((1 / 2) < 1 → (1 / 2) ≤ 1) |
20 | 14, 19 | ax-mp 5 | . . . . 5 ⊢ (1 / 2) ≤ 1 |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 / 2) ≤ 1) |
22 | 7, 12, 8, 13, 21 | letrd 11403 | . . 3 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ 1) |
23 | dnibndlem12.4 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) | |
24 | 3, 1, 23 | dnibndlem10 36093 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴)) |
25 | 9 | leabsd 15397 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (abs‘(𝐵 − 𝐴))) |
26 | 8, 9, 11, 24, 25 | letrd 11403 | . . 3 ⊢ (𝜑 → 1 ≤ (abs‘(𝐵 − 𝐴))) |
27 | 7, 8, 11, 22, 26 | letrd 11403 | . 2 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (abs‘(𝐵 − 𝐴))) |
28 | dnibndlem12.1 | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
29 | 28, 3, 1 | dnibndlem1 36084 | . 2 ⊢ (𝜑 → ((abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴)) ↔ (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (abs‘(𝐵 − 𝐴)))) |
30 | 27, 29 | mpbird 256 | 1 ⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℝcr 11139 1c1 11141 + caddc 11143 < clt 11280 ≤ cle 11281 − cmin 11476 / cdiv 11903 2c2 12300 ⌊cfl 13791 abscabs 15217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-fl 13793 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 |
This theorem is referenced by: dnibndlem13 36096 |
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