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Mirrors > Home > ILE Home > Th. List > modqlt | Unicode version |
Description: The modulo operation is less than its second argument. (Contributed by Jim Kingdon, 18-Oct-2021.) |
Ref | Expression |
---|---|
modqlt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9651 |
. . . . . 6
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2 | 1 | 3ad2ant1 1019 |
. . . . 5
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3 | qcn 9651 |
. . . . . 6
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4 | 3 | 3ad2ant2 1020 |
. . . . 5
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5 | qre 9642 |
. . . . . . 7
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6 | 5 | 3ad2ant2 1020 |
. . . . . 6
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7 | simp3 1000 |
. . . . . 6
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8 | 6, 7 | gt0ap0d 8603 |
. . . . 5
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9 | 2, 4, 8 | divcanap2d 8766 |
. . . 4
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10 | 9 | oveq1d 5905 |
. . 3
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11 | 7 | gt0ne0d 8486 |
. . . . . 6
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12 | qdivcl 9660 |
. . . . . 6
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13 | 11, 12 | syld3an3 1293 |
. . . . 5
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14 | qcn 9651 |
. . . . 5
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15 | 13, 14 | syl 14 |
. . . 4
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16 | 13 | flqcld 10294 |
. . . . 5
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17 | 16 | zcnd 9393 |
. . . 4
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18 | 4, 15, 17 | subdid 8388 |
. . 3
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19 | modqval 10341 |
. . 3
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20 | 10, 18, 19 | 3eqtr4rd 2232 |
. 2
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21 | qfraclt1 10297 |
. . . . 5
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22 | 13, 21 | syl 14 |
. . . 4
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23 | 4, 8 | dividapd 8760 |
. . . 4
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24 | 22, 23 | breqtrrd 4045 |
. . 3
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25 | qre 9642 |
. . . . . 6
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26 | 13, 25 | syl 14 |
. . . . 5
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27 | 16 | zred 9392 |
. . . . 5
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28 | 26, 27 | resubcld 8355 |
. . . 4
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29 | ltmuldiv2 8849 |
. . . 4
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30 | 28, 6, 6, 7, 29 | syl112anc 1252 |
. . 3
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31 | 24, 30 | mpbird 167 |
. 2
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32 | 20, 31 | eqbrtrd 4039 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 ax-pre-mulext 7946 ax-arch 7947 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-po 4310 df-iso 4311 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-div 8647 df-inn 8937 df-n0 9194 df-z 9271 df-q 9637 df-rp 9671 df-fl 10287 df-mod 10340 |
This theorem is referenced by: modqelico 10351 zmodfz 10363 modqid2 10368 modqabs 10374 modqmuladdim 10384 modaddmodup 10404 modqsubdir 10410 divalglemnn 11940 divalgmod 11949 bezoutlemnewy 12014 bezoutlemstep 12015 eucalglt 12074 odzdvds 12262 fldivp1 12363 4sqlem6 12398 lgseisenlem1 14833 |
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