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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 13697 | . . . . 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 12603 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℝ) |
5 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℝ) | |
6 | 5 | flcld 13695 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℤ) |
7 | 6 | peano2zd 12606 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℤ) |
8 | 7 | zred 12603 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → ((⌊‘𝐴) + 1) ∈ ℝ) |
9 | lelttr 11241 | . . . . 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 12550 | . . . 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 13696 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
16 | 15 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ≤ 𝐴) |
17 | 6 | zred 12603 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ) → (⌊‘𝐴) ∈ ℝ) |
18 | letr 11245 | . . . 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 5103 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 1c1 11048 + caddc 11050 < clt 11185 ≤ cle 11186 ℤcz 12495 ⌊cfl 13687 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
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 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 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-inf 9375 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-n0 12410 df-z 12496 df-uz 12760 df-fl 13689 |
This theorem is referenced by: fllt 13703 flid 13705 flwordi 13709 flval2 13711 flval3 13712 flge0nn0 13717 flge1nn 13718 flmulnn0 13724 btwnzge0 13725 fznnfl 13759 modmuladdnn0 13812 absrdbnd 15218 limsupgre 15355 climrlim2 15421 isprm7 16576 hashdvds 16639 prmreclem3 16782 ovolunlem1a 24844 mbfi1fseqlem4 25067 mbfi1fseqlem5 25068 dvfsumlem1 25374 dvfsumlem3 25376 ppisval 26437 dvdsflf1o 26520 ppiub 26536 chtub 26544 fsumvma2 26546 chpval2 26550 chpchtsum 26551 efexple 26613 bposlem3 26618 bposlem4 26619 bposlem5 26620 gausslemma2dlem4 26701 lgsquadlem1 26712 lgsquadlem2 26713 chebbnd1lem2 26802 chebbnd1lem3 26803 dchrisum0lem1 26848 pntrlog2bndlem6 26915 pntpbnd1 26918 pntpbnd2 26919 pntlemh 26931 pntlemj 26935 pntlemf 26937 aks4d1p1p2 40494 aks4d1p3 40502 aks4d1p6 40505 aks4d1p7d1 40506 aks4d1p7 40507 aks4d1p8 40511 aks4d1p9 40512 dirkertrigeqlem3 44273 nnolog2flm1 46608 |
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