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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 12688 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | halfre 12448 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
5 | 2, 4 | readdcld 11265 | . . 3 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
6 | dnizphlfeqhlf.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
7 | 6 | dnival 35882 | . . 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 11264 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | 4 | recnd 11264 | . . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ) |
11 | 9, 10 | addcld 11255 | . . . 4 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℂ) |
12 | 9, 10, 10 | addassd 11258 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + ((1 / 2) + (1 / 2)))) |
13 | 1cnd 11231 | . . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ) | |
14 | 13 | 2halvesd 12480 | . . . . . . . 8 ⊢ (𝜑 → ((1 / 2) + (1 / 2)) = 1) |
15 | 14 | oveq2d 7430 | . . . . . . 7 ⊢ (𝜑 → (𝐴 + ((1 / 2) + (1 / 2))) = (𝐴 + 1)) |
16 | 12, 15 | eqtrd 2767 | . . . . . 6 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) = (𝐴 + 1)) |
17 | 1 | peano2zd 12691 | . . . . . 6 ⊢ (𝜑 → (𝐴 + 1) ∈ ℤ) |
18 | 16, 17 | eqeltrd 2828 | . . . . 5 ⊢ (𝜑 → ((𝐴 + (1 / 2)) + (1 / 2)) ∈ ℤ) |
19 | flid 13797 | . . . . 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 11649 | . . 3 ⊢ (𝜑 → ((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2))) = (1 / 2)) |
22 | 21 | fveq2d 6895 | . 2 ⊢ (𝜑 → (abs‘((⌊‘((𝐴 + (1 / 2)) + (1 / 2))) − (𝐴 + (1 / 2)))) = (abs‘(1 / 2))) |
23 | halfgt0 12450 | . . . . 5 ⊢ 0 < (1 / 2) | |
24 | 0re 11238 | . . . . . 6 ⊢ 0 ∈ ℝ | |
25 | 24, 3 | ltlei 11358 | . . . . 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 15393 | . 2 ⊢ (𝜑 → (abs‘(1 / 2)) = (1 / 2)) |
29 | 8, 22, 28 | 3eqtrd 2771 | 1 ⊢ (𝜑 → (𝑇‘(𝐴 + (1 / 2))) = (1 / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 2c2 12289 ℤcz 12580 ⌊cfl 13779 abscabs 15205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fl 13781 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 |
This theorem is referenced by: knoppndvlem9 35931 |
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