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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 518 | . . 3 | |
2 | 1z 9087 | . . . 4 | |
3 | zq 9425 | . . . 4 | |
4 | 2, 3 | mp1i 10 | . . 3 |
5 | simprl 520 | . . 3 | |
6 | simprr 521 | . . 3 | |
7 | modqaddmod 10143 | . . 3 | |
8 | 1, 4, 5, 6, 7 | syl22anc 1217 | . 2 |
9 | 1, 5, 6 | modqcld 10108 | . . . 4 |
10 | qaddcl 9434 | . . . 4 | |
11 | 9, 4, 10 | syl2anc 408 | . . 3 |
12 | 0red 7774 | . . . 4 | |
13 | qre 9424 | . . . . 5 | |
14 | 9, 13 | syl 14 | . . . 4 |
15 | 1red 7788 | . . . . 5 | |
16 | 14, 15 | readdcld 7802 | . . . 4 |
17 | modqge0 10112 | . . . . 5 | |
18 | 1, 5, 6, 17 | syl3anc 1216 | . . . 4 |
19 | 14 | lep1d 8696 | . . . 4 |
20 | 12, 14, 16, 18, 19 | letrd 7893 | . . 3 |
21 | simplr 519 | . . . 4 | |
22 | qre 9424 | . . . . . 6 | |
23 | 5, 22 | syl 14 | . . . . 5 |
24 | 14, 15, 23 | ltaddsubd 8314 | . . . 4 |
25 | 21, 24 | mpbird 166 | . . 3 |
26 | modqid 10129 | . . 3 | |
27 | 11, 5, 20, 25, 26 | syl22anc 1217 | . 2 |
28 | 8, 27 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7626 cc0 7627 c1 7628 caddc 7630 clt 7807 cle 7808 cmin 7940 cz 9061 cq 9418 cmo 10102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 ax-arch 7746 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 df-inn 8728 df-n0 8985 df-z 9062 df-q 9419 df-rp 9449 df-fl 10050 df-mod 10103 |
This theorem is referenced by: (None) |
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