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Mirrors > Home > MPE Home > Th. List > Mathboxes > dnicld1 | Structured version Visualization version GIF version |
Description: Closure theorem for the "distance to nearest integer" function. (Contributed by Asger C. Ipsen, 4-Apr-2021.) |
Ref | Expression |
---|---|
dnicld1.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
dnicld1 | ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dnicld1.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | halfre 12229 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
4 | 1, 3 | jca 513 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ)) |
5 | readdcl 10996 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝐴 + (1 / 2)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
7 | reflcl 13558 | . . . . 5 ⊢ ((𝐴 + (1 / 2)) ∈ ℝ → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) | |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℝ) |
9 | 8 | recnd 11045 | . . 3 ⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈ ℂ) |
10 | 1 | recnd 11045 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | 9, 10 | subcld 11374 | . 2 ⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) − 𝐴) ∈ ℂ) |
12 | 11 | abscld 15189 | 1 ⊢ (𝜑 → (abs‘((⌊‘(𝐴 + (1 / 2))) − 𝐴)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2104 ‘cfv 6454 (class class class)co 7303 ℝcr 10912 1c1 10914 + caddc 10916 − cmin 11247 / cdiv 11674 2c2 12070 ⌊cfl 13552 abscabs 14986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-er 8525 df-en 8761 df-dom 8762 df-sdom 8763 df-sup 9241 df-inf 9242 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-n0 12276 df-z 12362 df-uz 12625 df-rp 12773 df-fl 13554 df-seq 13764 df-exp 13825 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 |
This theorem is referenced by: dnicld2 34694 dnif 34695 rddif2 34698 dnibndlem6 34704 dnibndlem7 34705 dnibndlem8 34706 dnibndlem9 34707 dnibndlem11 34709 dnibndlem12 34710 |
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