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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 12171 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℂ | |
2 | 1 | 2timesi 12094 | . . . . . . 7 ⊢ (2 · (1 / 2)) = ((1 / 2) + (1 / 2)) |
3 | 2cn 12031 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
4 | 2ne0 12060 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
5 | 3, 4 | recidi 11689 | . . . . . . 7 ⊢ (2 · (1 / 2)) = 1 |
6 | 2, 5 | eqtr3i 2769 | . . . . . 6 ⊢ ((1 / 2) + (1 / 2)) = 1 |
7 | 6 | oveq2i 7279 | . . . . 5 ⊢ ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = ((𝐴 − (1 / 2)) + 1) |
8 | recn 10945 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
9 | 1 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (1 / 2) ∈ ℂ) |
10 | 8, 9, 9 | nppcan3d 11342 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + ((1 / 2) + (1 / 2))) = (𝐴 + (1 / 2))) |
11 | 7, 10 | eqtr3id 2793 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) = (𝐴 + (1 / 2))) |
12 | halfre 12170 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
13 | readdcl 10938 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) | |
14 | 12, 13 | mpan2 687 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + (1 / 2)) ∈ ℝ) |
15 | fllep1 13502 | . . . . 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 5100 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1)) |
18 | resubcl 11268 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 − (1 / 2)) ∈ ℝ) | |
19 | 12, 18 | mpan2 687 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ∈ ℝ) |
20 | reflcl 13497 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
21 | 14, 20 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
22 | 1red 10960 | . . . 4 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ) | |
23 | 19, 21, 22 | leadd1d 11552 | . . 3 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2))) ↔ ((𝐴 − (1 / 2)) + 1) ≤ ((⌊‘(𝐴 + (1 / 2))) + 1))) |
24 | 17, 23 | mpbird 256 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − (1 / 2)) ≤ (⌊‘(𝐴 + (1 / 2)))) |
25 | flle 13500 | . . 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 15011 | . . 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 1369 | . 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 709 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ≤ (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2109 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 1c1 10856 + caddc 10858 · cmul 10860 ≤ cle 10994 − cmin 11188 / cdiv 11615 2c2 12011 ⌊cfl 13491 abscabs 14926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-rp 12713 df-fl 13493 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 |
This theorem is referenced by: absrdbnd 15034 rddif2 34636 dnibndlem11 34647 knoppcnlem4 34655 cntotbnd 35933 |
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