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Mirrors > Home > MPE Home > Th. List > flle | Structured version Visualization version GIF version |
Description: A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
flle | ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fllelt 13337 | . 2 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | |
2 | 1 | simpld 498 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 1c1 10695 + caddc 10697 < clt 10832 ≤ cle 10833 ⌊cfl 13330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fl 13332 |
This theorem is referenced by: fracge0 13344 flge 13345 flflp1 13347 flid 13348 flwordi 13352 flval2 13354 flval3 13355 fladdz 13365 flmulnn0 13367 fldiv4p1lem1div2 13375 fldiv4lem1div2uz2 13376 ceige 13383 flleceil 13391 fleqceilz 13392 quoremz 13393 quoremnn0ALT 13395 facavg 13832 rddif 14869 o1fsum 15340 flo1 15381 bitscmp 15960 isprm7 16228 prmreclem4 16435 zcld 23664 mbfi1fseqlem5 24571 mbfi1fseqlem6 24572 dvfsumlem1 24877 dvfsumlem2 24878 dvfsumlem3 24879 harmonicubnd 25846 harmonicbnd4 25847 ppisval 25940 ppiltx 26013 ppiub 26039 chtub 26047 chpub 26055 logfacubnd 26056 logfaclbnd 26057 bposlem1 26119 bposlem5 26123 bposlem6 26124 lgsquadlem1 26215 chebbnd1lem3 26306 vmadivsum 26317 dchrisumlem1 26324 dchrmusum2 26329 dchrisum0lem2a 26352 mudivsum 26365 mulogsumlem 26366 selberglem2 26381 selberg2lem 26385 pntrlog2bndlem4 26415 pntpbnd2 26422 pntlemg 26433 pntlemr 26437 pntlemk 26441 ostth2lem3 26470 dnibndlem4 34347 dnibndlem10 34353 knoppndvlem19 34396 ltflcei 35451 itg2addnclem3 35516 aks4d1p1p3 39759 aks4d1p1p2 39760 irrapxlem1 40288 hashnzfzclim 41554 fourierdlem4 43270 fourierdlem65 43330 fllogbd 45522 logbpw2m1 45529 fllog2 45530 nnpw2blen 45542 dignn0flhalflem2 45578 |
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