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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 11973 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | flle 13157 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ≤ 𝐴) |
4 | 1 | leidd 11194 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ 𝐴) |
5 | flge 13163 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℤ) → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) | |
6 | 1, 5 | mpancom 684 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) |
7 | 4, 6 | mpbid 233 | . 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ (⌊‘𝐴)) |
8 | reflcl 13154 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℝ) |
10 | 9, 1 | letri3d 10770 | . 2 ⊢ (𝐴 ∈ ℤ → ((⌊‘𝐴) = 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≤ (⌊‘𝐴)))) |
11 | 3, 7, 10 | mpbir2and 709 | 1 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 ℝcr 10524 ≤ cle 10664 ℤcz 11969 ⌊cfl 13148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-fl 13150 |
This theorem is referenced by: flidm 13167 flidz 13168 ceilid 13207 fleqceilz 13210 zmod10 13243 bits0 15765 bitsp1e 15769 bitsuz 15811 phiprmpw 16101 fldivp1 16221 prmreclem4 16243 dvfsumlem1 24550 dvfsumlem3 24552 ppival2 25632 ppival2g 25633 chtprm 25657 chtnprm 25658 chpp1 25659 chtdif 25662 cht1 25669 chp1 25671 prmorcht 25682 logfaclbnd 25725 logfacbnd3 25726 logexprlim 25728 rplogsumlem2 25988 log2sumbnd 26047 logdivbnd 26059 pntrsumbnd 26069 pntrlog2bndlem1 26080 pntrlog2bndlem4 26083 chpvalz 31798 chtvalz 31799 dnizphlfeqhlf 33712 lefldiveq 41435 fourierdlem65 42333 zefldiv2ALTV 43703 bits0ALTV 43721 zefldiv2 44518 flnn0div2ge 44521 flnn0ohalf 44522 nnlog2ge0lt1 44554 logbpw2m1 44555 blenpw2 44566 blen1 44572 blen2 44573 blengt1fldiv2p1 44581 dignn0fr 44589 dig0 44594 digexp 44595 0dig2nn0e 44600 0dig2nn0o 44601 |
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