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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem7 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33727. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem7.1 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
dnibndlem7 | ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibndlem7.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | halfre 11839 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
3 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
4 | 1, 3 | jca 512 | . . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ (1 / 2) ∈ ℝ)) |
5 | readdcl 10608 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐵 + (1 / 2)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
7 | reflcl 13154 | . . . . . 6 ⊢ ((𝐵 + (1 / 2)) ∈ ℝ → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℝ) |
9 | 8, 1 | jca 512 | . . . 4 ⊢ (𝜑 → ((⌊‘(𝐵 + (1 / 2))) ∈ ℝ ∧ 𝐵 ∈ ℝ)) |
10 | resubcl 10938 | . . . 4 ⊢ (((⌊‘(𝐵 + (1 / 2))) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((⌊‘(𝐵 + (1 / 2))) − 𝐵) ∈ ℝ) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → ((⌊‘(𝐵 + (1 / 2))) − 𝐵) ∈ ℝ) |
12 | 1 | dnicld1 33708 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) ∈ ℝ) |
13 | 11 | leabsd 14762 | . . 3 ⊢ (𝜑 → ((⌊‘(𝐵 + (1 / 2))) − 𝐵) ≤ (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
14 | 11, 12, 3, 13 | lesub2dd 11245 | . 2 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
15 | 3 | recnd 10657 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
16 | 8 | recnd 10657 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℂ) |
17 | 1 | recnd 10657 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | 15, 16, 17 | subsub3d 11015 | . . 3 ⊢ (𝜑 → ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵)) = (((1 / 2) + 𝐵) − (⌊‘(𝐵 + (1 / 2))))) |
19 | 15, 17 | addcomd 10830 | . . . 4 ⊢ (𝜑 → ((1 / 2) + 𝐵) = (𝐵 + (1 / 2))) |
20 | 19 | oveq1d 7160 | . . 3 ⊢ (𝜑 → (((1 / 2) + 𝐵) − (⌊‘(𝐵 + (1 / 2)))) = ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2))))) |
21 | 17, 16, 15 | subsub3d 11015 | . . . 4 ⊢ (𝜑 → (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) = ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2))))) |
22 | 21 | eqcomd 2824 | . . 3 ⊢ (𝜑 → ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2)))) = (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
23 | 18, 20, 22 | 3eqtrd 2857 | . 2 ⊢ (𝜑 → ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵)) = (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
24 | 14, 23 | breqtrd 5083 | 1 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 ≤ cle 10664 − cmin 10858 / cdiv 11285 2c2 11680 ⌊cfl 13148 abscabs 14581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fl 13150 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 |
This theorem is referenced by: dnibndlem9 33722 |
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