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Mirrors > Home > MPE Home > Th. List > flltp1 | Structured version Visualization version GIF version |
Description: A basic property of the floor (greatest integer) function. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
flltp1 | ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fllelt 13005 | . 2 ⊢ (𝐴 ∈ ℝ → ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 < ((⌊‘𝐴) + 1))) | |
2 | 1 | simprd 496 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 < ((⌊‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2079 class class class wbr 4956 ‘cfv 6217 (class class class)co 7007 ℝcr 10371 1c1 10373 + caddc 10375 < clt 10510 ≤ cle 10511 ⌊cfl 12998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-sup 8742 df-inf 8743 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-fl 13000 |
This theorem is referenced by: fllep1 13009 fraclt1 13010 flge 13013 flflp1 13015 fladdz 13033 flhalf 13038 ceim1l 13053 expnbnd 13431 efcllem 15252 bitscmp 15608 1arith 16080 zcld 23092 lebnumii 23241 lmnn 23537 vitalilem4 23883 bposlem1 25530 lgsquadlem1 25626 chebbnd1lem2 25716 dchrisumlem3 25737 pntrlog2bndlem2 25824 pntrlog2bndlem4 25826 pntlemh 25845 ostth2lem3 25881 minvecolem3 28332 dya2ub 31101 dnibndlem5 33374 ltflcei 34357 cntotbnd 34552 pellexlem5 38866 recnnltrp 41139 rpgtrecnn 41143 ioodvbdlimc1lem2 41712 ioodvbdlimc2lem 41714 fourierdlem4 41892 fourierdlem47 41934 fourierdlem65 41952 fllogbd 44055 nnpw2blen 44075 dignn0ldlem 44097 |
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