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Mirrors > Home > MPE Home > Th. List > flge0nn0 | Structured version Visualization version GIF version |
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 13762 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ) |
3 | 0z 12571 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | flge 13772 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) | |
5 | 3, 4 | mpan2 689 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) |
6 | 5 | biimpa 477 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (⌊‘𝐴)) |
7 | elnn0z 12573 | . 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 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6543 ℝcr 11111 0cc0 11112 ≤ cle 11251 ℕ0cn0 12474 ℤcz 12560 ⌊cfl 13757 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-n0 12475 df-z 12561 df-uz 12825 df-fl 13759 |
This theorem is referenced by: fldivnn0 13789 expnbnd 14197 facavg 14263 o1fsum 15761 efcllem 16023 odzdvds 16730 prmreclem3 16853 1arith 16862 odmodnn0 19410 lebnumii 24489 lmnn 24787 vitalilem4 25135 mbfi1fseqlem1 25240 mbfi1fseqlem3 25242 mbfi1fseqlem5 25244 harmoniclbnd 26520 harmonicbnd4 26522 fsumharmonic 26523 ppiltx 26688 logfac2 26727 chpval2 26728 chpchtsum 26729 chpub 26730 logfaclbnd 26732 logfacbnd3 26733 logfacrlim 26734 bposlem1 26794 gausslemma2dlem0d 26869 lgsquadlem2 26891 chtppilimlem1 26983 vmadivsum 26992 rpvmasumlem 26997 dchrisumlema 26998 dchrisumlem1 26999 dchrisum0lem1b 27025 dchrisum0lem1 27026 dchrisum0lem2a 27027 dchrisum0lem3 27029 mudivsum 27040 mulogsumlem 27041 selberglem2 27056 selberg2lem 27060 pntrsumo1 27075 pntrlog2bndlem2 27088 pntrlog2bndlem4 27090 pntrlog2bndlem6a 27092 pntpbnd1 27096 pntpbnd2 27097 pntlemg 27108 pntlemj 27113 pntlemf 27115 ostth2lem2 27144 ostth2lem3 27145 minvecolem3 30167 minvecolem4 30171 itg2addnclem2 36626 irrapxlem4 41645 irrapxlem5 41646 recnnltrp 44166 rpgtrecnn 44169 ioodvbdlimc1lem2 44727 ioodvbdlimc2lem 44729 fourierdlem47 44948 vonioolem1 45475 fllog2 47332 blennnelnn 47340 dignnld 47367 dignn0flhalf 47382 |
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