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Mirrors > Home > ILE Home > Th. List > summodnegmod | Unicode version |
Description: The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
Ref | Expression |
---|---|
summodnegmod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 940 | . . 3 | |
2 | simp1 938 | . . 3 | |
3 | znegcl 8382 | . . . 4 | |
4 | 3 | 3ad2ant2 960 | . . 3 |
5 | moddvds 10204 | . . 3 | |
6 | 1, 2, 4, 5 | syl3anc 1169 | . 2 |
7 | zcn 8356 | . . . . . 6 | |
8 | zcn 8356 | . . . . . 6 | |
9 | 7, 8 | anim12i 331 | . . . . 5 |
10 | 9 | 3adant3 958 | . . . 4 |
11 | subneg 7357 | . . . . 5 | |
12 | 11 | eqcomd 2086 | . . . 4 |
13 | 10, 12 | syl 14 | . . 3 |
14 | 13 | breq2d 3797 | . 2 |
15 | zaddcl 8391 | . . . 4 | |
16 | 15 | 3adant3 958 | . . 3 |
17 | dvdsval3 10199 | . . 3 | |
18 | 1, 16, 17 | syl2anc 403 | . 2 |
19 | 6, 14, 18 | 3bitr2rd 215 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 class class class wbr 3785 (class class class)co 5532 cc 6979 cc0 6981 caddc 6984 cmin 7279 cneg 7280 cn 8039 cz 8351 cmo 9324 cdvds 10195 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-q 8705 df-rp 8735 df-fl 9274 df-mod 9325 df-dvds 10196 |
This theorem is referenced by: (None) |
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