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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 13746 | . 2 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | |
2 | 1 | simpld 495 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5142 ‘cfv 6533 (class class class)co 7394 ℝcr 11093 1c1 11095 + caddc 11097 < clt 11232 ≤ cle 11233 ⌊cfl 13739 |
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 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 |
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 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-sup 9421 df-inf 9422 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-n0 12457 df-z 12543 df-uz 12807 df-fl 13741 |
This theorem is referenced by: fracge0 13753 flge 13754 flflp1 13756 flid 13757 flwordi 13761 flval2 13763 flval3 13764 fladdz 13774 flmulnn0 13776 fldiv4p1lem1div2 13784 fldiv4lem1div2uz2 13785 ceige 13793 flleceil 13802 fleqceilz 13803 quoremz 13804 quoremnn0ALT 13806 facavg 14245 rddif 15271 o1fsum 15743 flo1 15784 bitscmp 16363 isprm7 16629 prmreclem4 16836 zcld 24260 mbfi1fseqlem5 25168 mbfi1fseqlem6 25169 dvfsumlem1 25474 dvfsumlem2 25475 dvfsumlem3 25476 harmonicubnd 26443 harmonicbnd4 26444 ppisval 26537 ppiltx 26610 ppiub 26636 chtub 26644 chpub 26652 logfacubnd 26653 logfaclbnd 26654 bposlem1 26716 bposlem5 26720 bposlem6 26721 lgsquadlem1 26812 chebbnd1lem3 26903 vmadivsum 26914 dchrisumlem1 26921 dchrmusum2 26926 dchrisum0lem2a 26949 mudivsum 26962 mulogsumlem 26963 selberglem2 26978 selberg2lem 26982 pntrlog2bndlem4 27012 pntpbnd2 27019 pntlemg 27030 pntlemr 27034 pntlemk 27038 ostth2lem3 27067 dnibndlem4 35225 dnibndlem10 35231 knoppndvlem19 35274 ltflcei 36344 itg2addnclem3 36409 aks4d1p1p3 40803 aks4d1p1p2 40804 irrapxlem1 41395 hashnzfzclim 42916 fourierdlem4 44664 fourierdlem65 44724 fllogbd 46958 logbpw2m1 46965 fllog2 46966 nnpw2blen 46978 dignn0flhalflem2 47014 |
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