![]() |
Mathbox for Asger C. Ipsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dnibndlem8 | Structured version Visualization version GIF version |
Description: Lemma for dnibnd 33072. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnibndlem8.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
dnibndlem8 | ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnibndlem8.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | halfre 11601 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
4 | 1, 3 | jca 507 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ)) |
5 | simpl 476 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → 𝐴 ∈ ℝ) | |
6 | 2 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (1 / 2) ∈ ℝ) |
7 | 5, 6 | readdcld 10408 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) |
8 | 4, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
9 | reflcl 12921 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
11 | 1, 10 | resubcld 10806 | . . 3 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ∈ ℝ) |
12 | 1 | dnicld1 33053 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
13 | 11 | leabsd 14568 | . . . 4 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ≤ (abs‘(𝐴 − (⌊‘(𝐴 + (1 / 2)))))) |
14 | 1 | recnd 10407 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 10 | recnd 10407 | . . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℂ) |
16 | 14, 15 | abssubd 14607 | . . . 4 ⊢ (𝜑 → (abs‘(𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
17 | 13, 16 | breqtrd 4914 | . . 3 ⊢ (𝜑 → (𝐴 − (⌊‘(𝐴 + (1 / 2)))) ≤ (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
18 | 11, 12, 3, 17 | lesub2dd 10995 | . 2 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2)))))) |
19 | 3 | recnd 10407 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
20 | 19, 14, 15 | subsub3d 10766 | . . 3 ⊢ (𝜑 → ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) − 𝐴)) |
21 | 19, 15 | addcomd 10580 | . . . 4 ⊢ (𝜑 → ((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) = ((⌊‘(𝐴 + (1 / 2))) + (1 / 2))) |
22 | 21 | oveq1d 6939 | . . 3 ⊢ (𝜑 → (((1 / 2) + (⌊‘(𝐴 + (1 / 2)))) − 𝐴) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
23 | 20, 22 | eqtrd 2814 | . 2 ⊢ (𝜑 → ((1 / 2) − (𝐴 − (⌊‘(𝐴 + (1 / 2))))) = (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
24 | 18, 23 | breqtrd 4914 | 1 ⊢ (𝜑 → ((1 / 2) − (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) ≤ (((⌊‘(𝐴 + (1 / 2))) + (1 / 2)) − 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 1c1 10275 + caddc 10277 ≤ cle 10414 − cmin 10608 / cdiv 11035 2c2 11435 ⌊cfl 12915 abscabs 14387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-fl 12917 df-seq 13125 df-exp 13184 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 |
This theorem is referenced by: dnibndlem9 33067 |
Copyright terms: Public domain | W3C validator |