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| Description: The floor of a number greater than or equal to 0 is a nonnegative integer. (Contributed by NM, 26-Apr-2005.) | 
| Ref | Expression | 
|---|---|
| flge0nn0 | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | flcl 13836 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ) | 
| 3 | 0z 12626 | . . . 4 ⊢ 0 ∈ ℤ | |
| 4 | flge 13846 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) | |
| 5 | 3, 4 | mpan2 691 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) | 
| 6 | 5 | biimpa 476 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (⌊‘𝐴)) | 
| 7 | elnn0z 12628 | . 2 ⊢ ((⌊‘𝐴) ∈ ℕ0 ↔ ((⌊‘𝐴) ∈ ℤ ∧ 0 ≤ (⌊‘𝐴))) | |
| 8 | 2, 6, 7 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 ℝcr 11155 0cc0 11156 ≤ cle 11297 ℕ0cn0 12528 ℤcz 12615 ⌊cfl 13831 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fl 13833 | 
| This theorem is referenced by: fldivnn0 13863 expnbnd 14272 facavg 14341 o1fsum 15850 efcllem 16114 odzdvds 16834 prmreclem3 16957 1arith 16966 odmodnn0 19559 lebnumii 24999 lmnn 25298 vitalilem4 25647 mbfi1fseqlem1 25751 mbfi1fseqlem3 25753 mbfi1fseqlem5 25755 harmoniclbnd 27053 harmonicbnd4 27055 fsumharmonic 27056 ppiltx 27221 logfac2 27262 chpval2 27263 chpchtsum 27264 chpub 27265 logfaclbnd 27267 logfacbnd3 27268 logfacrlim 27269 bposlem1 27329 gausslemma2dlem0d 27404 lgsquadlem2 27426 chtppilimlem1 27518 vmadivsum 27527 rpvmasumlem 27532 dchrisumlema 27533 dchrisumlem1 27534 dchrisum0lem1b 27560 dchrisum0lem1 27561 dchrisum0lem2a 27562 dchrisum0lem3 27564 mudivsum 27575 mulogsumlem 27576 selberglem2 27591 selberg2lem 27595 pntrsumo1 27610 pntrlog2bndlem2 27623 pntrlog2bndlem4 27625 pntrlog2bndlem6a 27627 pntpbnd1 27631 pntpbnd2 27632 pntlemg 27643 pntlemj 27648 pntlemf 27650 ostth2lem2 27679 ostth2lem3 27680 minvecolem3 30896 minvecolem4 30900 itg2addnclem2 37680 irrapxlem4 42841 irrapxlem5 42842 recnnltrp 45393 rpgtrecnn 45396 ioodvbdlimc1lem2 45952 ioodvbdlimc2lem 45954 fourierdlem47 46173 vonioolem1 46700 fllog2 48494 blennnelnn 48502 dignnld 48529 dignn0flhalf 48544 | 
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