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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 964 |
. . . . . 6
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2 | nn0z 8978 |
. . . . . . 7
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3 | zq 9320 |
. . . . . . 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 272 |
. . . 4
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7 | nnq 9327 |
. . . . . 6
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8 | 7 | 3ad2ant2 986 |
. . . . 5
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9 | 8 | adantr 272 |
. . . 4
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10 | 1 | nn0ge0d 8937 |
. . . . 5
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11 | 10 | adantr 272 |
. . . 4
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12 | simpr 109 |
. . . 4
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13 | modqid 10015 |
. . . 4
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14 | 6, 9, 11, 12, 13 | syl22anc 1200 |
. . 3
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15 | iftrue 3445 |
. . . . 5
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16 | 15 | eqcomd 2120 |
. . . 4
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17 | 16 | adantl 273 |
. . 3
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18 | 14, 17 | eqtrd 2147 |
. 2
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19 | 5 | adantr 272 |
. . . 4
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20 | 8 | adantr 272 |
. . . 4
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21 | simp2 965 |
. . . . . 6
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22 | 21 | adantr 272 |
. . . . 5
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23 | 22 | nngt0d 8674 |
. . . 4
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24 | 21 | nnred 8643 |
. . . . . 6
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25 | 1 | nn0red 8935 |
. . . . . 6
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26 | 24, 25 | lenltd 7803 |
. . . . 5
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27 | 26 | biimpar 293 |
. . . 4
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28 | simpl3 969 |
. . . 4
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29 | q2submod 10051 |
. . . 4
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30 | 19, 20, 23, 27, 28, 29 | syl32anc 1207 |
. . 3
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31 | iffalse 3448 |
. . . . 5
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32 | 31 | adantl 273 |
. . . 4
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33 | 32 | eqcomd 2120 |
. . 3
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34 | 30, 33 | eqtrd 2147 |
. 2
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35 | 1, 2 | syl 14 |
. . 3
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36 | 21 | nnzd 9076 |
. . 3
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37 | zdclt 9032 |
. . . 4
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38 | exmiddc 804 |
. . . 4
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39 | 37, 38 | syl 14 |
. . 3
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40 | 35, 36, 39 | syl2anc 406 |
. 2
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41 | 18, 34, 40 | mpjaodan 770 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 ax-pre-mulext 7663 ax-arch 7664 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-po 4178 df-iso 4179 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-reap 8255 df-ap 8262 df-div 8346 df-inn 8631 df-2 8689 df-n0 8882 df-z 8959 df-q 9314 df-rp 9344 df-fl 9936 df-mod 9989 |
This theorem is referenced by: (None) |
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