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Mirrors > Home > ILE Home > Th. List > modqltm1p1mod | Unicode version |
Description: If a number modulo a modulus is less than the modulus decreased by 1, the first number increased by 1 modulo the modulus equals the first number modulo the modulus, increased by 1. (Contributed by Jim Kingdon, 24-Oct-2021.) |
Ref | Expression |
---|---|
modqltm1p1mod |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 497 |
. . 3
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2 | 1z 8839 |
. . . 4
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3 | zq 9174 |
. . . 4
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4 | 2, 3 | mp1i 10 |
. . 3
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5 | simprl 499 |
. . 3
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6 | simprr 500 |
. . 3
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7 | modqaddmod 9833 |
. . 3
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8 | 1, 4, 5, 6, 7 | syl22anc 1176 |
. 2
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9 | 1, 5, 6 | modqcld 9798 |
. . . 4
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10 | qaddcl 9183 |
. . . 4
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11 | 9, 4, 10 | syl2anc 404 |
. . 3
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12 | 0red 7552 |
. . . 4
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13 | qre 9173 |
. . . . 5
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14 | 9, 13 | syl 14 |
. . . 4
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15 | 1red 7566 |
. . . . 5
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16 | 14, 15 | readdcld 7580 |
. . . 4
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17 | modqge0 9802 |
. . . . 5
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18 | 1, 5, 6, 17 | syl3anc 1175 |
. . . 4
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19 | 14 | lep1d 8455 |
. . . 4
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20 | 12, 14, 16, 18, 19 | letrd 7670 |
. . 3
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21 | simplr 498 |
. . . 4
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22 | qre 9173 |
. . . . . 6
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23 | 5, 22 | syl 14 |
. . . . 5
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24 | 14, 15, 23 | ltaddsubd 8085 |
. . . 4
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25 | 21, 24 | mpbird 166 |
. . 3
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26 | modqid 9819 |
. . 3
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27 | 11, 5, 20, 25, 26 | syl22anc 1176 |
. 2
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28 | 8, 27 | eqtr3d 2123 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-id 4131 df-po 4134 df-iso 4135 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-n0 8737 df-z 8814 df-q 9168 df-rp 9198 df-fl 9740 df-mod 9793 |
This theorem is referenced by: (None) |
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