![]() |
Mathbox for Asger C. Ipsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dnizeq0 | Structured version Visualization version GIF version |
Description: The distance to nearest integer is zero for integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
Ref | Expression |
---|---|
dnizeq0.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
dnizeq0.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Ref | Expression |
---|---|
dnizeq0 | ⊢ (𝜑 → (𝑇‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnizeq0.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 1 | zred 12075 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | dnizeq0.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
4 | 3 | dnival 33923 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → (𝑇‘𝐴) = (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴))) |
6 | halfre 11839 | . . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
8 | 1, 7 | jca 515 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℤ ∧ (1 / 2) ∈ ℝ)) |
9 | flzadd 13191 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (1 / 2) ∈ ℝ) → (⌊‘(𝐴 + (1 / 2))) = (𝐴 + (⌊‘(1 / 2)))) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) = (𝐴 + (⌊‘(1 / 2)))) |
11 | 6 | rexri 10688 | . . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ* |
12 | 0re 10632 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ | |
13 | halfgt0 11841 | . . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2) | |
14 | 12, 6, 13 | ltleii 10752 | . . . . . . . . . . . . 13 ⊢ 0 ≤ (1 / 2) |
15 | halflt1 11843 | . . . . . . . . . . . . 13 ⊢ (1 / 2) < 1 | |
16 | 11, 14, 15 | 3pm3.2i 1336 | . . . . . . . . . . . 12 ⊢ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
17 | 0xr 10677 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ* | |
18 | 1xr 10689 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ* | |
19 | 17, 18 | pm3.2i 474 | . . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ* ∧ 1 ∈ ℝ*) |
20 | elico1 12769 | . . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → ((1 / 2) ∈ (0[,)1) ↔ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1))) | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ((1 / 2) ∈ (0[,)1) ↔ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1)) |
22 | 16, 21 | mpbir 234 | . . . . . . . . . . 11 ⊢ (1 / 2) ∈ (0[,)1) |
23 | 22 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 2) ∈ (0[,)1)) |
24 | ico01fl0 13184 | . . . . . . . . . 10 ⊢ ((1 / 2) ∈ (0[,)1) → (⌊‘(1 / 2)) = 0) | |
25 | 23, 24 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⌊‘(1 / 2)) = 0) |
26 | 25 | oveq2d 7151 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = (𝐴 + 0)) |
27 | 2 | recnd 10658 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
28 | 27 | addid1d 10829 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
29 | 26, 28 | eqtrd 2833 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = 𝐴) |
30 | 10, 29 | eqtrd 2833 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) = 𝐴) |
31 | 30 | oveq1d 7150 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = (𝐴 − 𝐴)) |
32 | 27 | subidd 10974 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
33 | 31, 32 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = 0) |
34 | 33 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = (abs‘0)) |
35 | abs0 14637 | . . . 4 ⊢ (abs‘0) = 0 | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → (abs‘0) = 0) |
37 | 34, 36 | eqtrd 2833 | . 2 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = 0) |
38 | 5, 37 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑇‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℤcz 11969 [,)cico 12728 ⌊cfl 13155 abscabs 14585 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: knoppndvlem6 33969 knoppndvlem8 33971 |
Copyright terms: Public domain | W3C validator |