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Mirrors > Home > ILE Home > Th. List > flqcl | Unicode version |
Description: The floor (greatest integer) function yields an integer when applied to a rational (closure law). For a similar closure law for real numbers apart from any integer, see flapcl 10305. (Contributed by Jim Kingdon, 8-Oct-2021.) |
Ref | Expression |
---|---|
flqcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qre 9654 |
. . 3
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2 | flval 10302 |
. . 3
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3 | 1, 2 | syl 14 |
. 2
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4 | qbtwnz 10281 |
. . 3
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5 | riotacl 5865 |
. . 3
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6 | 4, 5 | syl 14 |
. 2
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7 | 3, 6 | eqeltrd 2266 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 ax-arch 7959 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-inn 8949 df-n0 9206 df-z 9283 df-q 9649 df-rp 9683 df-fl 10300 |
This theorem is referenced by: flqlelt 10306 flqcld 10307 qfraclt1 10310 qfracge0 10311 flqidm 10315 flqidz 10316 flqge0nn0 10323 flqge1nn 10324 flqaddz 10327 flqzadd 10328 flqmulnn0 10329 ceilqval 10336 flqleceil 10347 flqeqceilz 10348 intqfrac2 10349 flqdiv 10351 modqfrac 10367 flqmod 10368 intqfrac 10369 modqmulnn 10372 |
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