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Mirrors > Home > MPE Home > Th. List > reflcl | Structured version Visualization version GIF version |
Description: The floor (greatest integer) function is real. (Contributed by NM, 15-Jul-2008.) |
Ref | Expression |
---|---|
reflcl | ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flcl 12790 | . 2 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
2 | 1 | zred 11674 | 1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 ‘cfv 6049 ℝcr 10127 ⌊cfl 12785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-fl 12787 |
This theorem is referenced by: fllep1 12796 fraclt1 12797 fracle1 12798 fracge0 12799 fllt 12801 flflp1 12802 flid 12803 flltnz 12806 flval3 12810 refldivcl 12818 fladdz 12820 flzadd 12821 flmulnn0 12822 flltdivnn0lt 12828 ceige 12838 ceim1l 12840 flleceil 12846 fleqceilz 12847 intfracq 12852 fldiv 12853 uzsup 12856 modvalr 12865 modfrac 12877 flmod 12878 intfrac 12879 modmulnn 12882 modcyc 12899 modadd1 12901 moddi 12932 modirr 12935 digit2 13191 digit1 13192 facavg 13282 rddif 14279 absrdbnd 14280 rexuzre 14291 o1fsum 14744 flo1 14785 isprm7 15622 opnmbllem 23569 mbfi1fseqlem1 23681 mbfi1fseqlem3 23683 mbfi1fseqlem4 23684 mbfi1fseqlem5 23685 mbfi1fseqlem6 23686 dvfsumlem1 23988 dvfsumlem2 23989 dvfsumlem3 23990 dvfsumlem4 23991 dvfsum2 23996 harmonicbnd4 24936 chtfl 25074 chpfl 25075 ppieq0 25101 ppiltx 25102 ppiub 25128 chpeq0 25132 chtub 25136 logfac2 25141 chpub 25144 logfacubnd 25145 logfaclbnd 25146 lgsquadlem1 25304 chtppilimlem1 25361 vmadivsum 25370 dchrisumlema 25376 dchrisumlem1 25377 dchrisumlem3 25379 dchrmusum2 25382 dchrisum0lem1b 25403 dchrisum0lem1 25404 dchrisum0lem2a 25405 dchrisum0lem3 25407 mudivsum 25418 mulogsumlem 25419 selberglem2 25434 pntrlog2bndlem6 25471 pntpbnd2 25475 pntlemg 25486 pntlemr 25490 pntlemj 25491 pntlemf 25493 pntlemk 25494 minvecolem4 28045 dnicld1 32768 dnibndlem2 32775 dnibndlem3 32776 dnibndlem4 32777 dnibndlem5 32778 dnibndlem7 32780 dnibndlem8 32781 dnibndlem9 32782 dnibndlem10 32783 dnibndlem11 32784 dnibndlem13 32786 dnibnd 32787 knoppcnlem4 32792 ltflcei 33710 leceifl 33711 opnmbllem0 33758 itg2addnclem2 33775 itg2addnclem3 33776 hashnzfzclim 39023 lefldiveq 40004 fourierdlem4 40831 fourierdlem26 40853 fourierdlem47 40873 fourierdlem65 40891 flsubz 42822 dignn0flhalflem2 42920 |
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