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Mirrors > Home > ILE Home > Th. List > flltdivnn0lt | Unicode version |
Description: The floor function of a division of a nonnegative integer by a positive integer is less than the division of a greater dividend by the same positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.) |
Ref | Expression |
---|---|
flltdivnn0lt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 998 |
. . . . . . 7
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2 | 1 | nn0zd 9386 |
. . . . . 6
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3 | simp3 1000 |
. . . . . 6
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4 | znq 9637 |
. . . . . . 7
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5 | 4 | flqcld 10290 |
. . . . . 6
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6 | 2, 3, 5 | syl2anc 411 |
. . . . 5
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7 | 6 | adantr 276 |
. . . 4
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8 | 7 | zred 9388 |
. . 3
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9 | 2 | adantr 276 |
. . . 4
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10 | 3 | adantr 276 |
. . . 4
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11 | qre 9638 |
. . . . 5
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12 | 4, 11 | syl 14 |
. . . 4
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13 | 9, 10, 12 | syl2anc 411 |
. . 3
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14 | simp2 999 |
. . . . . 6
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15 | 14 | nn0zd 9386 |
. . . . 5
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16 | 15 | adantr 276 |
. . . 4
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17 | znq 9637 |
. . . . 5
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18 | qre 9638 |
. . . . 5
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19 | 17, 18 | syl 14 |
. . . 4
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20 | 16, 10, 19 | syl2anc 411 |
. . 3
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21 | fldivnn0le 10316 |
. . . . 5
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22 | 21 | 3adant2 1017 |
. . . 4
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23 | 22 | adantr 276 |
. . 3
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24 | simpr 110 |
. . . 4
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25 | nn0re 9198 |
. . . . . . 7
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26 | nn0re 9198 |
. . . . . . 7
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27 | nnre 8939 |
. . . . . . . 8
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28 | nngt0 8957 |
. . . . . . . 8
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29 | 27, 28 | jca 306 |
. . . . . . 7
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30 | 25, 26, 29 | 3anim123i 1185 |
. . . . . 6
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31 | 30 | adantr 276 |
. . . . 5
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32 | ltdiv1 8838 |
. . . . 5
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33 | 31, 32 | syl 14 |
. . . 4
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34 | 24, 33 | mpbid 147 |
. . 3
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35 | 8, 13, 20, 23, 34 | lelttrd 8095 |
. 2
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36 | 35 | ex 115 |
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 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 |
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 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-po 4308 df-iso 4309 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-n0 9190 df-z 9267 df-q 9633 df-rp 9667 df-fl 10283 |
This theorem is referenced by: (None) |
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