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Mirrors > Home > MPE Home > Th. List > fllep1 | Structured version Visualization version GIF version |
Description: A basic property of the floor (greatest integer) function. (Contributed by Mario Carneiro, 21-May-2016.) |
Ref | Expression |
---|---|
fllep1 | ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flltp1 13805 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) | |
2 | reflcl 13801 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
3 | peano2re 11425 | . . . 4 ⊢ ((⌊‘𝐴) ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) + 1) ∈ ℝ) |
5 | ltle 11340 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ ((⌊‘𝐴) + 1) ∈ ℝ) → (𝐴 < ((⌊‘𝐴) + 1) → 𝐴 ≤ ((⌊‘𝐴) + 1))) | |
6 | 4, 5 | mpdan 685 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < ((⌊‘𝐴) + 1) → 𝐴 ≤ ((⌊‘𝐴) + 1))) |
7 | 1, 6 | mpd 15 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 ⌊cfl 13795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fl 13797 |
This theorem is referenced by: uzsup 13868 rddif 15327 rexuzre 15339 limsupgre 15465 rlimclim1 15529 o1fsum 15799 vdwnnlem3 16973 ovoliunlem2 25452 mbfi1fseqlem6 25670 dvfsumlem2 25981 dvfsumlem2OLD 25982 dvfsumlem3 25983 harmoniclbnd 26961 harmonicbnd4 26963 logfaclbnd 27175 chtppilimlem1 27426 dchrisumlema 27441 dchrisumlem3 27444 dchrisum0lem1 27469 selberg2lem 27503 pntrsumo1 27518 pntpbnd2 27540 pntlemg 27551 pntlemr 27555 pntlemj 27556 minvecolem4 30710 dstfrvunirn 34127 dnibndlem10 35995 knoppndvlem19 36038 ltflcei 37114 itg2addnclem3 37179 aks4d1p1p4 41574 aks6d1c7lem1 41684 irrapxlem4 42276 irrapxlem5 42277 |
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