Mathbox for Asger C. Ipsen |
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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 12090 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | dnizeq0.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
4 | 3 | dnival 33812 | . . 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 11854 | . . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
8 | 1, 7 | jca 514 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℤ ∧ (1 / 2) ∈ ℝ)) |
9 | flzadd 13199 | . . . . . . . 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 10701 | . . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ* |
12 | 0re 10645 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ | |
13 | halfgt0 11856 | . . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2) | |
14 | 12, 6, 13 | ltleii 10765 | . . . . . . . . . . . . 13 ⊢ 0 ≤ (1 / 2) |
15 | halflt1 11858 | . . . . . . . . . . . . 13 ⊢ (1 / 2) < 1 | |
16 | 11, 14, 15 | 3pm3.2i 1335 | . . . . . . . . . . . 12 ⊢ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
17 | 0xr 10690 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ* | |
18 | 1xr 10702 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ* | |
19 | 17, 18 | pm3.2i 473 | . . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ* ∧ 1 ∈ ℝ*) |
20 | elico1 12784 | . . . . . . . . . . . . 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 233 | . . . . . . . . . . 11 ⊢ (1 / 2) ∈ (0[,)1) |
23 | 22 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 2) ∈ (0[,)1)) |
24 | ico01fl0 13192 | . . . . . . . . . 10 ⊢ ((1 / 2) ∈ (0[,)1) → (⌊‘(1 / 2)) = 0) | |
25 | 23, 24 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⌊‘(1 / 2)) = 0) |
26 | 25 | oveq2d 7174 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = (𝐴 + 0)) |
27 | 2 | recnd 10671 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
28 | 27 | addid1d 10842 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
29 | 26, 28 | eqtrd 2858 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = 𝐴) |
30 | 10, 29 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) = 𝐴) |
31 | 30 | oveq1d 7173 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = (𝐴 − 𝐴)) |
32 | 27 | subidd 10987 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
33 | 31, 32 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = 0) |
34 | 33 | fveq2d 6676 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = (abs‘0)) |
35 | abs0 14647 | . . . 4 ⊢ (abs‘0) = 0 | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → (abs‘0) = 0) |
37 | 34, 36 | eqtrd 2858 | . 2 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = 0) |
38 | 5, 37 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝑇‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = 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 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 2c2 11695 ℤcz 11984 [,)cico 12743 ⌊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-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 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: knoppndvlem6 33858 knoppndvlem8 33860 |
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