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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 12508 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
2 | flle 13710 | . . 3 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ≤ 𝐴) |
4 | 1 | leidd 11726 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ≤ 𝐴) |
5 | flge 13716 | . . . 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 13707 | . . . 4 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ) | |
9 | 1, 8 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℤ → (⌊‘𝐴) ∈ ℝ) |
10 | 9, 1 | letri3d 11302 | . 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 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 ℝcr 11055 ≤ cle 11195 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: flidm 13720 flidz 13721 ceilid 13762 fleqceilz 13765 zmod10 13798 bits0 16313 bitsp1e 16317 bitsuz 16359 phiprmpw 16653 fldivp1 16774 prmreclem4 16796 dvfsumlem1 25406 dvfsumlem3 25408 ppival2 26493 ppival2g 26494 chtprm 26518 chtnprm 26519 chpp1 26520 chtdif 26523 cht1 26530 chp1 26532 prmorcht 26543 logfaclbnd 26586 logfacbnd3 26587 logexprlim 26589 rplogsumlem2 26849 log2sumbnd 26908 logdivbnd 26920 pntrsumbnd 26930 pntrlog2bndlem1 26941 pntrlog2bndlem4 26944 chpvalz 33298 chtvalz 33299 dnizphlfeqhlf 34985 lefldiveq 43613 fourierdlem65 44498 zefldiv2ALTV 45939 bits0ALTV 45957 zefldiv2 46702 flnn0div2ge 46705 flnn0ohalf 46706 nnlog2ge0lt1 46738 logbpw2m1 46739 blenpw2 46750 blen1 46756 blen2 46757 blengt1fldiv2p1 46765 dignn0fr 46773 dig0 46778 digexp 46779 0dig2nn0e 46784 0dig2nn0o 46785 |
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