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Mirrors > Home > MPE Home > Th. List > flo1 | Structured version Visualization version GIF version |
Description: The floor function satisfies ⌊(𝑥) = 𝑥 + 𝑂(1). (Contributed by Mario Carneiro, 21-May-2016.) |
Ref | Expression |
---|---|
flo1 | ⊢ (𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 4001 | . . 3 ⊢ (⊤ → ℝ ⊆ ℝ) | |
2 | reflcl 13787 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ) | |
3 | resubcl 11548 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ (⌊‘𝑥) ∈ ℝ) → (𝑥 − (⌊‘𝑥)) ∈ ℝ) | |
4 | 2, 3 | mpdan 686 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑥 − (⌊‘𝑥)) ∈ ℝ) |
5 | 4 | recnd 11266 | . . . 4 ⊢ (𝑥 ∈ ℝ → (𝑥 − (⌊‘𝑥)) ∈ ℂ) |
6 | 5 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (𝑥 − (⌊‘𝑥)) ∈ ℂ) |
7 | 1red 11239 | . . 3 ⊢ (⊤ → 1 ∈ ℝ) | |
8 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ) | |
9 | flle 13790 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥) | |
10 | 2, 8, 9 | abssubge0d 15404 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (abs‘(𝑥 − (⌊‘𝑥))) = (𝑥 − (⌊‘𝑥))) |
11 | fracle1 13794 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑥 − (⌊‘𝑥)) ≤ 1) | |
12 | 10, 11 | eqbrtrd 5164 | . . . 4 ⊢ (𝑥 ∈ ℝ → (abs‘(𝑥 − (⌊‘𝑥))) ≤ 1) |
13 | 12 | ad2antrl 727 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)) → (abs‘(𝑥 − (⌊‘𝑥))) ≤ 1) |
14 | 1, 6, 7, 7, 13 | elo1d 15506 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1)) |
15 | 14 | mptru 1541 | 1 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 − (⌊‘𝑥))) ∈ 𝑂(1) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1535 ∈ wcel 2099 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 1c1 11133 ≤ cle 11273 − cmin 11468 ⌊cfl 13781 abscabs 15207 𝑂(1)co1 15456 |
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 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-rp 13001 df-ico 13356 df-fl 13783 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-o1 15460 df-lo1 15461 |
This theorem is referenced by: (None) |
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