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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem7 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 34892. (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 12325 | . . . . . . . . 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 11092 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐵 + (1 / 2)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
7 | reflcl 13655 | . . . . . 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 11423 | . . . 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 34873 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵)) ∈ ℝ) |
13 | 11 | leabsd 15259 | . . 3 ⊢ (𝜑 → ((⌊‘(𝐵 + (1 / 2))) − 𝐵) ≤ (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
14 | 11, 12, 3, 13 | lesub2dd 11730 | . 2 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐵 + (1 / 2))) − 𝐵))) ≤ ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵))) |
15 | 3 | recnd 11141 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
16 | 8 | recnd 11141 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈ ℂ) |
17 | 1 | recnd 11141 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
18 | 15, 16, 17 | subsub3d 11500 | . . 3 ⊢ (𝜑 → ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵)) = (((1 / 2) + 𝐵) − (⌊‘(𝐵 + (1 / 2))))) |
19 | 15, 17 | addcomd 11315 | . . . 4 ⊢ (𝜑 → ((1 / 2) + 𝐵) = (𝐵 + (1 / 2))) |
20 | 19 | oveq1d 7366 | . . 3 ⊢ (𝜑 → (((1 / 2) + 𝐵) − (⌊‘(𝐵 + (1 / 2)))) = ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2))))) |
21 | 17, 16, 15 | subsub3d 11500 | . . . 4 ⊢ (𝜑 → (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2))) = ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2))))) |
22 | 21 | eqcomd 2743 | . . 3 ⊢ (𝜑 → ((𝐵 + (1 / 2)) − (⌊‘(𝐵 + (1 / 2)))) = (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
23 | 18, 20, 22 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → ((1 / 2) − ((⌊‘(𝐵 + (1 / 2))) − 𝐵)) = (𝐵 − ((⌊‘(𝐵 + (1 / 2))) − (1 / 2)))) |
24 | 14, 23 | breqtrd 5129 | 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 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7351 ℝcr 11008 1c1 11010 + caddc 11012 ≤ cle 11148 − cmin 11343 / cdiv 11770 2c2 12166 ⌊cfl 13649 abscabs 15079 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-fl 13651 df-seq 13861 df-exp 13922 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 |
This theorem is referenced by: dnibndlem9 34887 |
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