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Mirrors > Home > MPE Home > Th. List > flge | Structured version Visualization version GIF version |
Description: The floor function value is the greatest integer less than or equal to its argument. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
Ref | Expression |
---|---|
flge | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flltp1 13761 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | |
2 | 1 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐴 < ((⌊‘𝐴) + 1)) |
3 | simpr 485 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ) | |
4 | 3 | zred 12662 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ) |
5 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ) | |
6 | 5 | flcld 13759 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ) |
7 | 6 | peano2zd 12665 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℤ) |
8 | 7 | zred 12662 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ) |
9 | lelttr 11300 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((⌊‘𝐴) + 1) ∈ ℝ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)) → 𝐵 < ((⌊‘𝐴) + 1))) | |
10 | 4, 5, 8, 9 | syl3anc 1371 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1)) → 𝐵 < ((⌊‘𝐴) + 1))) |
11 | 2, 10 | mpan2d 692 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → 𝐵 < ((⌊‘𝐴) + 1))) |
12 | zleltp1 12609 | . . . 4 ⊢ ((𝐵 ∈ ℤ ∧ (⌊‘𝐴) ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) ↔ 𝐵 < ((⌊‘𝐴) + 1))) | |
13 | 3, 6, 12 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) ↔ 𝐵 < ((⌊‘𝐴) + 1))) |
14 | 11, 13 | sylibrd 258 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 → 𝐵 ≤ (⌊‘𝐴))) |
15 | flle 13760 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴) |
17 | 6 | zred 12662 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ) |
18 | letr 11304 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ (⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 ≤ (⌊‘𝐴) ∧ (⌊‘𝐴) ≤ 𝐴) → 𝐵 ≤ 𝐴)) | |
19 | 4, 17, 5, 18 | syl3anc 1371 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((𝐵 ≤ (⌊‘𝐴) ∧ (⌊‘𝐴) ≤ 𝐴) → 𝐵 ≤ 𝐴)) |
20 | 16, 19 | mpan2d 692 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ (⌊‘𝐴) → 𝐵 ≤ 𝐴)) |
21 | 14, 20 | impbid 211 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (𝐵 ≤ 𝐴 ↔ 𝐵 ≤ (⌊‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 ℤcz 12554 ⌊cfl 13751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fl 13753 |
This theorem is referenced by: fllt 13767 flid 13769 flwordi 13773 flval2 13775 flval3 13776 flge0nn0 13781 flge1nn 13782 flmulnn0 13788 btwnzge0 13789 fznnfl 13823 modmuladdnn0 13876 absrdbnd 15284 limsupgre 15421 climrlim2 15487 isprm7 16641 hashdvds 16704 prmreclem3 16847 ovolunlem1a 25004 mbfi1fseqlem4 25227 mbfi1fseqlem5 25228 dvfsumlem1 25534 dvfsumlem3 25536 ppisval 26597 dvdsflf1o 26680 ppiub 26696 chtub 26704 fsumvma2 26706 chpval2 26710 chpchtsum 26711 efexple 26773 bposlem3 26778 bposlem4 26779 bposlem5 26780 gausslemma2dlem4 26861 lgsquadlem1 26872 lgsquadlem2 26873 chebbnd1lem2 26962 chebbnd1lem3 26963 dchrisum0lem1 27008 pntrlog2bndlem6 27075 pntpbnd1 27078 pntpbnd2 27079 pntlemh 27091 pntlemj 27095 pntlemf 27097 aks4d1p1p2 40923 aks4d1p3 40931 aks4d1p6 40934 aks4d1p7d1 40935 aks4d1p7 40936 aks4d1p8 40940 aks4d1p9 40941 dirkertrigeqlem3 44802 nnolog2flm1 47229 |
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