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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem12 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33943. (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 33924 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) ∈ ℝ) |
3 | dnibndlem12.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 3 | dnicld1 33924 | . . . . . 6 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
5 | 2, 4 | resubcld 11057 | . . . . 5 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℝ) |
6 | 5 | recnd 10658 | . . . 4 ⊢ (𝜑 → ((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ∈ ℂ) |
7 | 6 | abscld 14788 | . . 3 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ∈ ℝ) |
8 | 1red 10631 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
9 | 1, 3 | resubcld 11057 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
10 | 9 | recnd 10658 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
11 | 10 | abscld 14788 | . . 3 ⊢ (𝜑 → (abs‘(𝐵 − 𝐴)) ∈ ℝ) |
12 | 8 | rehalfcld 11872 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
13 | 3, 1 | dnibndlem11 33940 | . . . 4 ⊢ (𝜑 → (abs‘((abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)))) ≤ (1 / 2)) |
14 | halflt1 11843 | . . . . . 6 ⊢ (1 / 2) < 1 | |
15 | halfre 11839 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
16 | 1re 10630 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
17 | 15, 16 | pm3.2i 474 | . . . . . . 7 ⊢ ((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) |
18 | ltle 10718 | . . . . . . 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 10786 | . . 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 33939 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝐵 − 𝐴)) |
25 | 9 | leabsd 14766 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (abs‘(𝐵 − 𝐴))) |
26 | 8, 9, 11, 24, 25 | letrd 10786 | . . 3 ⊢ (𝜑 → 1 ≤ (abs‘(𝐵 − 𝐴))) |
27 | 7, 8, 11, 22, 26 | letrd 10786 | . 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 33930 | . 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 260 | 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 < clt 10664 ≤ 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: dnibndlem13 33942 |
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