![]() |
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 11834 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | dnizeq0.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
4 | 3 | dnival 33044 | . . 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 11596 | . . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
8 | 1, 7 | jca 507 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℤ ∧ (1 / 2) ∈ ℝ)) |
9 | flzadd 12946 | . . . . . . . 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 10435 | . . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ* |
12 | 0re 10378 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ | |
13 | halfgt0 11598 | . . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2) | |
14 | 12, 6, 13 | ltleii 10499 | . . . . . . . . . . . . 13 ⊢ 0 ≤ (1 / 2) |
15 | halflt1 11600 | . . . . . . . . . . . . 13 ⊢ (1 / 2) < 1 | |
16 | 11, 14, 15 | 3pm3.2i 1395 | . . . . . . . . . . . 12 ⊢ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
17 | 0xr 10423 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ* | |
18 | 1xr 10436 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ* | |
19 | 17, 18 | pm3.2i 464 | . . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ* ∧ 1 ∈ ℝ*) |
20 | elico1 12530 | . . . . . . . . . . . . 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 223 | . . . . . . . . . . 11 ⊢ (1 / 2) ∈ (0[,)1) |
23 | 22 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 2) ∈ (0[,)1)) |
24 | ico01fl0 12939 | . . . . . . . . . 10 ⊢ ((1 / 2) ∈ (0[,)1) → (⌊‘(1 / 2)) = 0) | |
25 | 23, 24 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⌊‘(1 / 2)) = 0) |
26 | 25 | oveq2d 6938 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = (𝐴 + 0)) |
27 | 2 | recnd 10405 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
28 | 27 | addid1d 10576 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
29 | 26, 28 | eqtrd 2813 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = 𝐴) |
30 | 10, 29 | eqtrd 2813 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) = 𝐴) |
31 | 30 | oveq1d 6937 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = (𝐴 − 𝐴)) |
32 | 27 | subidd 10722 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
33 | 31, 32 | eqtrd 2813 | . . . 4 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = 0) |
34 | 33 | fveq2d 6450 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = (abs‘0)) |
35 | abs0 14432 | . . . 4 ⊢ (abs‘0) = 0 | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → (abs‘0) = 0) |
37 | 34, 36 | eqtrd 2813 | . 2 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = 0) |
38 | 5, 37 | eqtrd 2813 | 1 ⊢ (𝜑 → (𝑇‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 2c2 11430 ℤcz 11728 [,)cico 12489 ⌊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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 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-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-ico 12493 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: knoppndvlem6 33090 knoppndvlem8 33092 |
Copyright terms: Public domain | W3C validator |