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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 13764 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
2 | 1 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ) |
3 | 0z 12573 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | flge 13774 | . . . 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 12575 | . 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 11253 ℕ0cn0 12476 ℤcz 12562 ⌊cfl 13759 |
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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fl 13761 |
This theorem is referenced by: fldivnn0 13791 expnbnd 14199 facavg 14265 o1fsum 15763 efcllem 16025 odzdvds 16732 prmreclem3 16855 1arith 16864 odmodnn0 19449 lebnumii 24706 lmnn 25004 vitalilem4 25352 mbfi1fseqlem1 25457 mbfi1fseqlem3 25459 mbfi1fseqlem5 25461 harmoniclbnd 26737 harmonicbnd4 26739 fsumharmonic 26740 ppiltx 26905 logfac2 26944 chpval2 26945 chpchtsum 26946 chpub 26947 logfaclbnd 26949 logfacbnd3 26950 logfacrlim 26951 bposlem1 27011 gausslemma2dlem0d 27086 lgsquadlem2 27108 chtppilimlem1 27200 vmadivsum 27209 rpvmasumlem 27214 dchrisumlema 27215 dchrisumlem1 27216 dchrisum0lem1b 27242 dchrisum0lem1 27243 dchrisum0lem2a 27244 dchrisum0lem3 27246 mudivsum 27257 mulogsumlem 27258 selberglem2 27273 selberg2lem 27277 pntrsumo1 27292 pntrlog2bndlem2 27305 pntrlog2bndlem4 27307 pntrlog2bndlem6a 27309 pntpbnd1 27313 pntpbnd2 27314 pntlemg 27325 pntlemj 27330 pntlemf 27332 ostth2lem2 27361 ostth2lem3 27362 minvecolem3 30384 minvecolem4 30388 itg2addnclem2 36843 irrapxlem4 41865 irrapxlem5 41866 recnnltrp 44386 rpgtrecnn 44389 ioodvbdlimc1lem2 44947 ioodvbdlimc2lem 44949 fourierdlem47 45168 vonioolem1 45695 fllog2 47342 blennnelnn 47350 dignnld 47377 dignn0flhalf 47392 |
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