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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 13706 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ) | |
2 | 1 | adantr 482 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ) |
3 | 0z 12515 | . . . 4 ⊢ 0 ∈ ℤ | |
4 | flge 13716 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) | |
5 | 3, 4 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ 0 ≤ (⌊‘𝐴))) |
6 | 5 | biimpa 478 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 0 ≤ (⌊‘𝐴)) |
7 | elnn0z 12517 | . 2 ⊢ ((⌊‘𝐴) ∈ ℕ0 ↔ ((⌊‘𝐴) ∈ ℤ ∧ 0 ≤ (⌊‘𝐴))) | |
8 | 2, 6, 7 | sylanbrc 584 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 ℝcr 11055 0cc0 11056 ≤ cle 11195 ℕ0cn0 12418 ℤcz 12504 ⌊cfl 13701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fl 13703 |
This theorem is referenced by: fldivnn0 13733 expnbnd 14141 facavg 14207 o1fsum 15703 efcllem 15965 odzdvds 16672 prmreclem3 16795 1arith 16804 odmodnn0 19327 lebnumii 24345 lmnn 24643 vitalilem4 24991 mbfi1fseqlem1 25096 mbfi1fseqlem3 25098 mbfi1fseqlem5 25100 harmoniclbnd 26374 harmonicbnd4 26376 fsumharmonic 26377 ppiltx 26542 logfac2 26581 chpval2 26582 chpchtsum 26583 chpub 26584 logfaclbnd 26586 logfacbnd3 26587 logfacrlim 26588 bposlem1 26648 gausslemma2dlem0d 26723 lgsquadlem2 26745 chtppilimlem1 26837 vmadivsum 26846 rpvmasumlem 26851 dchrisumlema 26852 dchrisumlem1 26853 dchrisum0lem1b 26879 dchrisum0lem1 26880 dchrisum0lem2a 26881 dchrisum0lem3 26883 mudivsum 26894 mulogsumlem 26895 selberglem2 26910 selberg2lem 26914 pntrsumo1 26929 pntrlog2bndlem2 26942 pntrlog2bndlem4 26944 pntrlog2bndlem6a 26946 pntpbnd1 26950 pntpbnd2 26951 pntlemg 26962 pntlemj 26967 pntlemf 26969 ostth2lem2 26998 ostth2lem3 26999 minvecolem3 29860 minvecolem4 29864 itg2addnclem2 36176 irrapxlem4 41191 irrapxlem5 41192 recnnltrp 43698 rpgtrecnn 43701 ioodvbdlimc1lem2 44259 ioodvbdlimc2lem 44261 fourierdlem47 44480 vonioolem1 45007 fllog2 46740 blennnelnn 46748 dignnld 46775 dignn0flhalf 46790 |
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