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Mirrors > Home > MPE Home > Th. List > rddif | Structured version Visualization version GIF version |
Description: The difference between a real number and its nearest integer is less than or equal to one half. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Sep-2015.) |
Ref | Expression |
---|---|
rddif | ⊢ (𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfcn 11597 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℂ | |
2 | 1 | 2timesi 11520 | . . . . . . 7 ⊢ (2 · (1 / 2)) = ((1 / 2) + (1 / 2)) |
3 | 2cn 11450 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
4 | 2ne0 11486 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
5 | 3, 4 | recidi 11106 | . . . . . . 7 ⊢ (2 · (1 / 2)) = 1 |
6 | 2, 5 | eqtr3i 2804 | . . . . . 6 ⊢ ((1 / 2) + (1 / 2)) = 1 |
7 | 6 | oveq2i 6933 | . . . . 5 ⊢ ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = ((𝐴 − (1 / 2)) + 1) |
8 | recn 10362 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℂ) |
10 | 8, 9, 9 | nppcan3d 10761 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = (𝐴 + (1 / 2))) |
11 | 7, 10 | syl5eqr 2828 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) = (𝐴 + (1 / 2))) |
12 | halfre 11596 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
13 | readdcl 10355 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) | |
14 | 12, 13 | mpan2 681 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) ∈ ℝ) |
15 | fllep1 12921 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (𝐴 + (1 / 2)) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1)) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1)) |
17 | 11, 16 | eqbrtrd 4908 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1)) |
18 | resubcl 10687 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 − (1 / 2)) ∈ ℝ) | |
19 | 12, 18 | mpan2 681 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ∈ ℝ) |
20 | reflcl 12916 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
21 | 14, 20 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
22 | 1red 10377 | . . . 4 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ) | |
23 | 19, 21, 22 | leadd1d 10969 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2))) ↔ ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1))) |
24 | 17, 23 | mpbird 249 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2)))) |
25 | flle 12919 | . . 3 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ≤ (𝐴 + (1 / 2))) | |
26 | 14, 25 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ≤ (𝐴 + (1 / 2))) |
27 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
28 | 12 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℝ) |
29 | absdifle 14465 | . . 3 ⊢ (((⌊‘(𝐴 + (1 / 2))) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → ((abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2) ↔ ((𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2))) ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (𝐴 + (1 / 2))))) | |
30 | 21, 27, 28, 29 | syl3anc 1439 | . 2 ⊢ (𝐴 ∈ ℝ → ((abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2) ↔ ((𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2))) ∧ (⌊‘(𝐴 + (1 / 2))) ≤ (𝐴 + (1 / 2))))) |
31 | 24, 26, 30 | mpbir2and 703 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 1c1 10273 + caddc 10275 · cmul 10277 ≤ cle 10412 − cmin 10606 / cdiv 11032 2c2 11430 ⌊cfl 12910 abscabs 14381 |
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 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
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 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fl 12912 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 |
This theorem is referenced by: absrdbnd 14488 rddif2 33050 dnibndlem11 33061 knoppcnlem4 33069 cntotbnd 34221 |
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