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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 12586 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | flle 13790 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ≤ 𝐴) |
4 | 1 | leidd 11804 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ 𝐴) |
5 | flge 13796 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℤ) → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) | |
6 | 1, 5 | mpancom 687 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ≤ 𝐴 ↔ 𝐴 ≤ (⌊‘𝐴))) |
7 | 4, 6 | mpbid 231 | . 2 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ (⌊‘𝐴)) |
8 | reflcl 13787 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℝ) |
10 | 9, 1 | letri3d 11380 | . 2 ⊢ (𝐴 ∈ ℤ → ((⌊‘𝐴) = 𝐴 ↔ ((⌊‘𝐴) ≤ 𝐴 ∧ 𝐴 ≤ (⌊‘𝐴)))) |
11 | 3, 7, 10 | mpbir2and 712 | 1 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ‘cfv 6542 ℝcr 11131 ≤ cle 11273 ℤcz 12582 ⌊cfl 13781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-n0 12497 df-z 12583 df-uz 12847 df-fl 13783 |
This theorem is referenced by: flidm 13800 flidz 13801 ceilid 13842 fleqceilz 13845 zmod10 13878 bits0 16396 bitsp1e 16400 bitsuz 16442 phiprmpw 16738 fldivp1 16859 prmreclem4 16881 dvfsumlem1 25953 dvfsumlem3 25956 ppival2 27053 ppival2g 27054 chtprm 27078 chtnprm 27079 chpp1 27080 chtdif 27083 cht1 27090 chp1 27092 prmorcht 27103 logfaclbnd 27148 logfacbnd3 27149 logexprlim 27151 rplogsumlem2 27411 log2sumbnd 27470 logdivbnd 27482 pntrsumbnd 27492 pntrlog2bndlem1 27503 pntrlog2bndlem4 27506 chpvalz 34254 chtvalz 34255 dnizphlfeqhlf 35945 lefldiveq 44668 fourierdlem65 45553 zefldiv2ALTV 46995 bits0ALTV 47013 zefldiv2 47597 flnn0div2ge 47600 flnn0ohalf 47601 nnlog2ge0lt1 47633 logbpw2m1 47634 blenpw2 47645 blen1 47651 blen2 47652 blengt1fldiv2p1 47660 dignn0fr 47668 dig0 47673 digexp 47674 0dig2nn0e 47679 0dig2nn0o 47680 |
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