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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 11846 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℂ | |
2 | 1 | 2timesi 11769 | . . . . . . 7 ⊢ (2 · (1 / 2)) = ((1 / 2) + (1 / 2)) |
3 | 2cn 11706 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
4 | 2ne0 11735 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
5 | 3, 4 | recidi 11365 | . . . . . . 7 ⊢ (2 · (1 / 2)) = 1 |
6 | 2, 5 | eqtr3i 2846 | . . . . . 6 ⊢ ((1 / 2) + (1 / 2)) = 1 |
7 | 6 | oveq2i 7161 | . . . . 5 ⊢ ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = ((𝐴 − (1 / 2)) + 1) |
8 | recn 10621 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℂ) |
10 | 8, 9, 9 | nppcan3d 11018 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = (𝐴 + (1 / 2))) |
11 | 7, 10 | syl5eqr 2870 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) = (𝐴 + (1 / 2))) |
12 | halfre 11845 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
13 | readdcl 10614 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) | |
14 | 12, 13 | mpan2 689 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) ∈ ℝ) |
15 | fllep1 13165 | . . . . 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 5080 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1)) |
18 | resubcl 10944 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 − (1 / 2)) ∈ ℝ) | |
19 | 12, 18 | mpan2 689 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ∈ ℝ) |
20 | reflcl 13160 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
21 | 14, 20 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
22 | 1red 10636 | . . . 4 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ) | |
23 | 19, 21, 22 | leadd1d 11228 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2))) ↔ ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1))) |
24 | 17, 23 | mpbird 259 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2)))) |
25 | flle 13163 | . . 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 14672 | . . 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 1367 | . 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 711 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 ≤ cle 10670 − cmin 10864 / cdiv 11291 2c2 11686 ⌊cfl 13154 abscabs 14587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13156 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 |
This theorem is referenced by: absrdbnd 14695 rddif2 33811 dnibndlem11 33822 knoppcnlem4 33830 cntotbnd 35068 |
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