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Mirrors > Home > ILE Home > Th. List > modifeq2int | Unicode version |
Description: If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
Ref | Expression |
---|---|
modifeq2int |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 999 |
. . . . . 6
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2 | nn0z 9291 |
. . . . . . 7
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3 | zq 9644 |
. . . . . . 7
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4 | 2, 3 | syl 14 |
. . . . . 6
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5 | 1, 4 | syl 14 |
. . . . 5
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6 | 5 | adantr 276 |
. . . 4
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7 | nnq 9651 |
. . . . . 6
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8 | 7 | 3ad2ant2 1021 |
. . . . 5
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9 | 8 | adantr 276 |
. . . 4
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10 | 1 | nn0ge0d 9250 |
. . . . 5
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11 | 10 | adantr 276 |
. . . 4
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12 | simpr 110 |
. . . 4
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13 | modqid 10367 |
. . . 4
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14 | 6, 9, 11, 12, 13 | syl22anc 1250 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | iftrue 3554 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | eqcomd 2195 |
. . . 4
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17 | 16 | adantl 277 |
. . 3
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18 | 14, 17 | eqtrd 2222 |
. 2
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19 | 5 | adantr 276 |
. . . 4
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20 | 8 | adantr 276 |
. . . 4
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21 | simp2 1000 |
. . . . . 6
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22 | 21 | adantr 276 |
. . . . 5
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23 | 22 | nngt0d 8981 |
. . . 4
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24 | 21 | nnred 8950 |
. . . . . 6
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25 | 1 | nn0red 9248 |
. . . . . 6
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26 | 24, 25 | lenltd 8093 |
. . . . 5
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27 | 26 | biimpar 297 |
. . . 4
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28 | simpl3 1004 |
. . . 4
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29 | q2submod 10403 |
. . . 4
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30 | 19, 20, 23, 27, 28, 29 | syl32anc 1257 |
. . 3
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31 | iffalse 3557 |
. . . . 5
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32 | 31 | adantl 277 |
. . . 4
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33 | 32 | eqcomd 2195 |
. . 3
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34 | 30, 33 | eqtrd 2222 |
. 2
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35 | 1, 2 | syl 14 |
. . 3
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36 | 21 | nnzd 9392 |
. . 3
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37 | zdclt 9348 |
. . . 4
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38 | exmiddc 837 |
. . . 4
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39 | 37, 38 | syl 14 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 35, 36, 39 | syl2anc 411 |
. 2
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41 | 18, 34, 40 | mpjaodan 799 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 |
This theorem depends on definitions: df-bi 117 df-dc 836 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-if 3550 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 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-n0 9195 df-z 9272 df-q 9638 df-rp 9672 df-fl 10288 df-mod 10341 |
This theorem is referenced by: (None) |
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