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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnizphlfeqhlf | Structured version Visualization version GIF version |
Description: The distance to nearest integer is a half for half-integers. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
Ref | Expression |
---|---|
dnizphlfeqhlf.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
dnizphlfeqhlf.1 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
Ref | Expression |
---|---|
dnizphlfeqhlf | ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnizphlfeqhlf.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
2 | 1 | zred 11899 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | halfre 11660 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
5 | 2, 4 | readdcld 10468 | . . 3 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
6 | dnizphlfeqhlf.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
7 | 6 | dnival 33363 | . . 3 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (𝑇‘(𝐴 + (1 / 2))) = (abs‘((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2))))) |
8 | 5, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (abs‘((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2))))) |
9 | 2 | recnd 10467 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | 4 | recnd 10467 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
11 | 9, 10 | addcld 10458 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℂ) |
12 | 9, 10, 10 | addassd 10461 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + ((1 / 2) + (1 / 2)))) |
13 | 1cnd 10433 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 13 | 2halvesd 11692 | . . . . . . . 8 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1) |
15 | 14 | oveq2d 6991 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + ((1 / 2) + (1 / 2))) = (𝐴 + 1)) |
16 | 12, 15 | eqtrd 2809 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + 1)) |
17 | 1 | peano2zd 11902 | . . . . . 6 ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
18 | 16, 17 | eqeltrd 2861 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) ∈ ℤ) |
19 | flid 12992 | . . . . 5 ⊢ (((𝐴 + (1 / 2)) + (1 / 2)) ∈ ℤ → (⌊‘((𝐴 + (1 / 2)) + (1 / 2))) = ((𝐴 + (1 / 2)) + (1 / 2))) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → (⌊‘((𝐴 + (1 / 2)) + (1 / 2))) = ((𝐴 + (1 / 2)) + (1 / 2))) |
21 | 11, 10, 20 | mvrladdd 10853 | . . 3 ⊢ (𝜑 → ((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2))) = (1 / 2)) |
22 | 21 | fveq2d 6501 | . 2 ⊢ (𝜑 → (abs‘((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2)))) = (abs‘(1 / 2))) |
23 | halfgt0 11662 | . . . . 5 ⊢ 0 < (1 / 2) | |
24 | 0re 10440 | . . . . . 6 ⊢ 0 ∈ ℝ | |
25 | 24, 3 | ltlei 10561 | . . . . 5 ⊢ (0 < (1 / 2) → 0 ≤ (1 / 2)) |
26 | 23, 25 | ax-mp 5 | . . . 4 ⊢ 0 ≤ (1 / 2) |
27 | 26 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ≤ (1 / 2)) |
28 | 4, 27 | absidd 14642 | . 2 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2)) |
29 | 8, 22, 28 | 3eqtrd 2813 | 1 ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 class class class wbr 4926 ↦ cmpt 5005 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 0cc0 10334 1c1 10335 + caddc 10337 < clt 10473 ≤ cle 10474 − cmin 10669 / cdiv 11097 2c2 11494 ℤcz 11792 ⌊cfl 12974 abscabs 14453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 df-uz 12058 df-rp 12204 df-fl 12976 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 |
This theorem is referenced by: knoppndvlem9 33412 |
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