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Mirrors > Home > ILE Home > Th. List > flqcld | Unicode version |
Description: The floor (greatest integer) function is an integer (closure law). (Contributed by Jim Kingdon, 8-Oct-2021.) |
Ref | Expression |
---|---|
flqcld.1 |
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Ref | Expression |
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flqcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flqcld.1 |
. 2
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2 | flqcl 10276 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-mulrcl 7913 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-precex 7924 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-apti 7929 ax-pre-ltadd 7930 ax-pre-mulgt0 7931 ax-pre-mulext 7932 ax-arch 7933 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-reap 8535 df-ap 8542 df-div 8633 df-inn 8923 df-n0 9180 df-z 9257 df-q 9623 df-rp 9657 df-fl 10273 |
This theorem is referenced by: flqge 10285 flqlt 10286 flid 10287 flqltnz 10290 flqwordi 10291 flqword2 10292 flqaddz 10300 flhalf 10305 flltdivnn0lt 10307 fldiv4p1lem1div2 10308 ceiqcl 10310 ceiqge 10312 ceiqm1l 10314 intfracq 10323 flqdiv 10324 modqval 10327 modqvalr 10328 modqcl 10329 flqpmodeq 10330 modq0 10332 modqge0 10335 modqlt 10336 modqdiffl 10338 modqdifz 10339 modqmulnn 10345 modqvalp1 10346 zmodcl 10347 modqcyc 10362 modqadd1 10364 modqmuladd 10369 modqmul1 10380 modqdi 10395 modqsubdir 10396 iexpcyc 10628 facavg 10729 dvdsmod 11871 divalglemnn 11926 divalgmod 11935 flodddiv4t2lthalf 11945 modgcd 11995 hashdvds 12224 prmdiv 12238 odzdvds 12248 fldivp1 12349 pcfac 12351 pcbc 12352 mulgmodid 13036 |
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