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Mirrors > Home > ILE Home > Th. List > flqeqceilz | Unicode version |
Description: A rational number is an integer iff its floor equals its ceiling. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Ref | Expression |
---|---|
flqeqceilz | ⌈ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flid 9126 | . . 3 | |
2 | ceilid 9157 | . . 3 ⌈ | |
3 | 1, 2 | eqtr4d 2075 | . 2 ⌈ |
4 | flqcl 9117 | . . . . . 6 | |
5 | zq 8561 | . . . . . 6 | |
6 | 4, 5 | syl 14 | . . . . 5 |
7 | qdceq 9102 | . . . . 5 DECID | |
8 | 6, 7 | mpancom 399 | . . . 4 DECID |
9 | exmiddc 744 | . . . 4 DECID | |
10 | 8, 9 | syl 14 | . . 3 |
11 | eqeq1 2046 | . . . . . . 7 ⌈ ⌈ | |
12 | 11 | adantr 261 | . . . . . 6 ⌈ ⌈ |
13 | ceilqidz 9158 | . . . . . . . . 9 ⌈ | |
14 | eqcom 2042 | . . . . . . . . 9 ⌈ ⌈ | |
15 | 13, 14 | syl6bb 185 | . . . . . . . 8 ⌈ |
16 | 15 | biimprd 147 | . . . . . . 7 ⌈ |
17 | 16 | adantl 262 | . . . . . 6 ⌈ |
18 | 12, 17 | sylbid 139 | . . . . 5 ⌈ |
19 | 18 | ex 108 | . . . 4 ⌈ |
20 | flqle 9120 | . . . . 5 | |
21 | df-ne 2206 | . . . . . 6 | |
22 | necom 2289 | . . . . . . 7 | |
23 | qltlen 8575 | . . . . . . . . . . 11 | |
24 | 6, 23 | mpancom 399 | . . . . . . . . . 10 |
25 | breq1 3767 | . . . . . . . . . . . . . 14 ⌈ ⌈ | |
26 | 25 | adantl 262 | . . . . . . . . . . . . 13 ⌈ ⌈ |
27 | ceilqge 9152 | . . . . . . . . . . . . . . 15 ⌈ | |
28 | qre 8560 | . . . . . . . . . . . . . . . . 17 | |
29 | ceilqcl 9150 | . . . . . . . . . . . . . . . . . 18 ⌈ | |
30 | 29 | zred 8360 | . . . . . . . . . . . . . . . . 17 ⌈ |
31 | 28, 30 | lenltd 7134 | . . . . . . . . . . . . . . . 16 ⌈ ⌈ |
32 | pm2.21 547 | . . . . . . . . . . . . . . . 16 ⌈ ⌈ | |
33 | 31, 32 | syl6bi 152 | . . . . . . . . . . . . . . 15 ⌈ ⌈ |
34 | 27, 33 | mpd 13 | . . . . . . . . . . . . . 14 ⌈ |
35 | 34 | adantr 261 | . . . . . . . . . . . . 13 ⌈ ⌈ |
36 | 26, 35 | sylbid 139 | . . . . . . . . . . . 12 ⌈ |
37 | 36 | ex 108 | . . . . . . . . . . 11 ⌈ |
38 | 37 | com23 72 | . . . . . . . . . 10 ⌈ |
39 | 24, 38 | sylbird 159 | . . . . . . . . 9 ⌈ |
40 | 39 | expd 245 | . . . . . . . 8 ⌈ |
41 | 40 | com3r 73 | . . . . . . 7 ⌈ |
42 | 22, 41 | sylbi 114 | . . . . . 6 ⌈ |
43 | 21, 42 | sylbir 125 | . . . . 5 ⌈ |
44 | 20, 43 | mpdi 38 | . . . 4 ⌈ |
45 | 19, 44 | jaoi 636 | . . 3 ⌈ |
46 | 10, 45 | mpcom 32 | . 2 ⌈ |
47 | 3, 46 | impbid2 131 | 1 ⌈ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 DECID wdc 742 wceq 1243 wcel 1393 wne 2204 class class class wbr 3764 cfv 4902 clt 7060 cle 7061 cz 8245 cq 8554 cfl 9112 ⌈cceil 9113 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002 ax-arch 7003 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-n0 8182 df-z 8246 df-q 8555 df-rp 8584 df-fl 9114 df-ceil 9115 |
This theorem is referenced by: (None) |
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