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Mirrors > Home > ILE Home > Th. List > modqadd1 | Unicode version |
Description: Addition property of the modulo operation. (Contributed by Jim Kingdon, 22-Oct-2021.) |
Ref | Expression |
---|---|
modqadd1.a |
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modqadd1.b |
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modqadd1.c |
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modqadd1.dq |
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modqadd1.dgt0 |
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modqadd1.ab |
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Ref | Expression |
---|---|
modqadd1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modqadd1.ab |
. 2
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2 | modqadd1.a |
. . . . . . 7
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3 | modqadd1.dq |
. . . . . . 7
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4 | modqadd1.dgt0 |
. . . . . . 7
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5 | modqval 9727 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 2, 3, 4, 5 | syl3anc 1174 |
. . . . . 6
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7 | modqadd1.b |
. . . . . . 7
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8 | modqval 9727 |
. . . . . . 7
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9 | 7, 3, 4, 8 | syl3anc 1174 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 6, 9 | eqeq12d 2102 |
. . . . 5
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11 | oveq1 5659 |
. . . . 5
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12 | 10, 11 | syl6bi 161 |
. . . 4
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13 | qcn 9117 |
. . . . . . 7
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14 | 2, 13 | syl 14 |
. . . . . 6
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15 | modqadd1.c |
. . . . . . 7
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16 | qcn 9117 |
. . . . . . 7
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17 | 15, 16 | syl 14 |
. . . . . 6
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18 | qcn 9117 |
. . . . . . . 8
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19 | 3, 18 | syl 14 |
. . . . . . 7
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20 | 4 | gt0ne0d 7988 |
. . . . . . . . . 10
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21 | qdivcl 9126 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 2, 3, 20, 21 | syl3anc 1174 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | flqcld 9680 |
. . . . . . . 8
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24 | 23 | zcnd 8867 |
. . . . . . 7
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25 | 19, 24 | mulcld 7506 |
. . . . . 6
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26 | 14, 17, 25 | addsubd 7812 |
. . . . 5
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27 | qcn 9117 |
. . . . . . 7
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28 | 7, 27 | syl 14 |
. . . . . 6
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29 | qdivcl 9126 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 7, 3, 20, 29 | syl3anc 1174 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 30 | flqcld 9680 |
. . . . . . . 8
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32 | 31 | zcnd 8867 |
. . . . . . 7
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33 | 19, 32 | mulcld 7506 |
. . . . . 6
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34 | 28, 17, 33 | addsubd 7812 |
. . . . 5
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35 | 26, 34 | eqeq12d 2102 |
. . . 4
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36 | 12, 35 | sylibrd 167 |
. . 3
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37 | oveq1 5659 |
. . . 4
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38 | qaddcl 9118 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | 2, 15, 38 | syl2anc 403 |
. . . . . 6
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40 | modqcyc2 9763 |
. . . . . 6
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41 | 39, 23, 3, 4, 40 | syl22anc 1175 |
. . . . 5
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42 | qaddcl 9118 |
. . . . . . 7
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43 | 7, 15, 42 | syl2anc 403 |
. . . . . 6
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44 | modqcyc2 9763 |
. . . . . 6
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45 | 43, 31, 3, 4, 44 | syl22anc 1175 |
. . . . 5
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46 | 41, 45 | eqeq12d 2102 |
. . . 4
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47 | 37, 46 | syl5ib 152 |
. . 3
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48 | 36, 47 | syld 44 |
. 2
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49 | 1, 48 | mpd 13 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 ax-arch 7462 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-n0 8672 df-z 8749 df-q 9103 df-rp 9133 df-fl 9673 df-mod 9726 |
This theorem is referenced by: modqaddabs 9765 modqaddmod 9766 modqadd12d 9783 modqaddmulmod 9794 moddvds 11079 |
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