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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 12460 | . 2 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | |
2 | 1 | simpld 474 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ⌊cfl 12453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fl 12455 |
This theorem is referenced by: fracge0 12467 flge 12468 flflp1 12470 flid 12471 flwordi 12475 flval2 12477 flval3 12478 fladdz 12488 flmulnn0 12490 fldiv4p1lem1div2 12498 fldiv4lem1div2uz2 12499 ceige 12506 flleceil 12514 fleqceilz 12515 quoremz 12516 quoremnn0ALT 12518 facavg 12950 rddif 13928 o1fsum 14386 flo1 14425 bitscmp 14998 isprm7 15258 prmreclem4 15461 zcld 22424 mbfi1fseqlem5 23292 mbfi1fseqlem6 23293 dvfsumlem1 23593 dvfsumlem2 23594 dvfsumlem3 23595 harmonicubnd 24536 harmonicbnd4 24537 ppisval 24630 ppiltx 24703 ppiub 24729 chtub 24737 chpub 24745 logfacubnd 24746 logfaclbnd 24747 bposlem1 24809 bposlem5 24813 bposlem6 24814 lgsquadlem1 24905 chebbnd1lem3 24960 vmadivsum 24971 dchrisumlem1 24978 dchrmusum2 24983 dchrisum0lem2a 25006 mudivsum 25019 mulogsumlem 25020 selberglem2 25035 selberg2lem 25039 pntrlog2bndlem4 25069 pntpbnd2 25076 pntlemg 25087 pntlemr 25091 pntlemk 25095 ostth2lem3 25124 dnibndlem4 31641 dnibndlem10 31647 knoppndvlem19 31691 ltflcei 32567 itg2addnclem3 32633 irrapxlem1 36404 hashnzfzclim 37543 fourierdlem4 39004 fourierdlem65 39064 fllogbd 42152 logbpw2m1 42159 fllog2 42160 nnpw2blen 42172 dignn0flhalflem2 42208 |
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