Mathbox for Asger C. Ipsen |
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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 12090 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | halfre 11854 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
5 | 2, 4 | readdcld 10672 | . . 3 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
6 | dnizphlfeqhlf.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
7 | 6 | dnival 33812 | . . 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 10671 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | 4 | recnd 10671 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
11 | 9, 10 | addcld 10662 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℂ) |
12 | 9, 10, 10 | addassd 10665 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + ((1 / 2) + (1 / 2)))) |
13 | 1cnd 10638 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 13 | 2halvesd 11886 | . . . . . . . 8 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1) |
15 | 14 | oveq2d 7174 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + ((1 / 2) + (1 / 2))) = (𝐴 + 1)) |
16 | 12, 15 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + 1)) |
17 | 1 | peano2zd 12093 | . . . . . 6 ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
18 | 16, 17 | eqeltrd 2915 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) ∈ ℤ) |
19 | flid 13181 | . . . . 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 11055 | . . 3 ⊢ (𝜑 → ((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2))) = (1 / 2)) |
22 | 21 | fveq2d 6676 | . 2 ⊢ (𝜑 → (abs‘((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2)))) = (abs‘(1 / 2))) |
23 | halfgt0 11856 | . . . . 5 ⊢ 0 < (1 / 2) | |
24 | 0re 10645 | . . . . . 6 ⊢ 0 ∈ ℝ | |
25 | 24, 3 | ltlei 10764 | . . . . 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 14784 | . 2 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2)) |
29 | 8, 22, 28 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 2c2 11695 ℤcz 11984 ⌊cfl 13163 abscabs 14595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: knoppndvlem9 33861 |
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