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Mirrors > Home > MPE Home > Th. List > flid | Structured version Visualization version GIF version |
Description: An integer is its own floor. (Contributed by NM, 15-Nov-2004.) |
Ref | Expression |
---|---|
flid | ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 11583 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | flle 12808 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ≤ 𝐴) |
4 | 1 | leidd 10796 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ 𝐴) |
5 | flge 12814 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℤ) → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) | |
6 | 1, 5 | mpancom 668 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) |
7 | 4, 6 | mpbid 222 | . 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ (⌊‘𝐴)) |
8 | reflcl 12805 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℝ) |
10 | 9, 1 | letri3d 10381 | . 2 ⊢ (𝐴 ∈ ℤ → ((⌊‘𝐴) = 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≤ (⌊‘𝐴)))) |
11 | 3, 7, 10 | mpbir2and 692 | 1 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 ℝcr 10137 ≤ cle 10277 ℤcz 11579 ⌊cfl 12799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-inf 8505 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 df-z 11580 df-uz 11889 df-fl 12801 |
This theorem is referenced by: flidm 12818 flidz 12819 ceilid 12858 fleqceilz 12861 zmod10 12894 bits0 15358 bitsp1e 15362 bitsuz 15404 phiprmpw 15688 fldivp1 15808 prmreclem4 15830 dvfsumlem1 24009 dvfsumlem3 24011 ppival2 25075 ppival2g 25076 chtprm 25100 chtnprm 25101 chpp1 25102 chtdif 25105 cht1 25112 chp1 25114 prmorcht 25125 logfaclbnd 25168 logfacbnd3 25169 logexprlim 25171 rplogsumlem2 25395 log2sumbnd 25454 logdivbnd 25466 pntrsumbnd 25476 pntrlog2bndlem1 25487 pntrlog2bndlem4 25490 chpvalz 31046 chtvalz 31047 dnizphlfeqhlf 32803 lefldiveq 40023 fourierdlem65 40905 zefldiv2ALTV 42101 bits0ALTV 42118 zefldiv2 42852 flnn0div2ge 42855 flnn0ohalf 42856 nnlog2ge0lt1 42888 logbpw2m1 42889 blenpw2 42900 blen1 42906 blen2 42907 blengt1fldiv2p1 42915 dignn0fr 42923 dig0 42928 digexp 42929 0dig2nn0e 42934 0dig2nn0o 42935 |
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