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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 12720 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | dnizeq0.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
4 | 3 | dnival 36454 | . . 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 12478 | . . . . . . . . . 10 ⊢ (1 / 2) ∈ ℝ | |
7 | 6 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
8 | 1, 7 | jca 511 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℤ ∧ (1 / 2) ∈ ℝ)) |
9 | flzadd 13863 | . . . . . . . 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 11317 | . . . . . . . . . . . . 13 ⊢ (1 / 2) ∈ ℝ* |
12 | 0re 11261 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ | |
13 | halfgt0 12480 | . . . . . . . . . . . . . 14 ⊢ 0 < (1 / 2) | |
14 | 12, 6, 13 | ltleii 11382 | . . . . . . . . . . . . 13 ⊢ 0 ≤ (1 / 2) |
15 | halflt1 12482 | . . . . . . . . . . . . 13 ⊢ (1 / 2) < 1 | |
16 | 11, 14, 15 | 3pm3.2i 1338 | . . . . . . . . . . . 12 ⊢ ((1 / 2) ∈ ℝ* ∧ 0 ≤ (1 / 2) ∧ (1 / 2) < 1) |
17 | 0xr 11306 | . . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ* | |
18 | 1xr 11318 | . . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ* | |
19 | 17, 18 | pm3.2i 470 | . . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ* ∧ 1 ∈ ℝ*) |
20 | elico1 13427 | . . . . . . . . . . . . 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 231 | . . . . . . . . . . 11 ⊢ (1 / 2) ∈ (0[,)1) |
23 | 22 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (1 / 2) ∈ (0[,)1)) |
24 | ico01fl0 13856 | . . . . . . . . . 10 ⊢ ((1 / 2) ∈ (0[,)1) → (⌊‘(1 / 2)) = 0) | |
25 | 23, 24 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (⌊‘(1 / 2)) = 0) |
26 | 25 | oveq2d 7447 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = (𝐴 + 0)) |
27 | 2 | recnd 11287 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
28 | 27 | addridd 11459 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
29 | 26, 28 | eqtrd 2775 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + (⌊‘(1 / 2))) = 𝐴) |
30 | 10, 29 | eqtrd 2775 | . . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) = 𝐴) |
31 | 30 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = (𝐴 − 𝐴)) |
32 | 27 | subidd 11606 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
33 | 31, 32 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) = 0) |
34 | 33 | fveq2d 6911 | . . 3 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = (abs‘0)) |
35 | abs0 15321 | . . . 4 ⊢ (abs‘0) = 0 | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → (abs‘0) = 0) |
37 | 34, 36 | eqtrd 2775 | . 2 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) = 0) |
38 | 5, 37 | eqtrd 2775 | 1 ⊢ (𝜑 → (𝑇‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 2c2 12319 ℤcz 12611 [,)cico 13386 ⌊cfl 13827 abscabs 15270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-fl 13829 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: knoppndvlem6 36500 knoppndvlem8 36502 |
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