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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1001 |
. . 3
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2 | simp1 999 |
. . 3
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3 | znegcl 9314 |
. . . 4
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4 | 3 | 3ad2ant2 1021 |
. . 3
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5 | moddvds 11838 |
. . 3
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6 | 1, 2, 4, 5 | syl3anc 1249 |
. 2
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7 | zcn 9288 |
. . . . . 6
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8 | zcn 9288 |
. . . . . 6
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9 | 7, 8 | anim12i 338 |
. . . . 5
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10 | 9 | 3adant3 1019 |
. . . 4
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11 | subneg 8236 |
. . . . 5
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12 | 11 | eqcomd 2195 |
. . . 4
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13 | 10, 12 | syl 14 |
. . 3
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14 | 13 | breq2d 4030 |
. 2
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15 | zaddcl 9323 |
. . . 4
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16 | 15 | 3adant3 1019 |
. . 3
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17 | dvdsval3 11830 |
. . 3
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18 | 1, 16, 17 | syl2anc 411 |
. 2
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19 | 6, 14, 18 | 3bitr2rd 217 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-n0 9207 df-z 9284 df-q 9650 df-rp 9684 df-fl 10301 df-mod 10354 df-dvds 11827 |
This theorem is referenced by: (None) |
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