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Mirrors > Home > ILE Home > Th. List > zmodid2 | Unicode version |
Description: Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
zmodid2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zq 9677 |
. . . 4
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2 | 1 | adantr 276 |
. . 3
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3 | nnq 9684 |
. . . 4
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4 | 3 | adantl 277 |
. . 3
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5 | nngt0 8993 |
. . . 4
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6 | 5 | adantl 277 |
. . 3
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7 | modqid2 10408 |
. . 3
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8 | 2, 4, 6, 7 | syl3anc 1249 |
. 2
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9 | nnz 9322 |
. . . 4
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10 | 0z 9314 |
. . . . . 6
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11 | elfzm11 10143 |
. . . . . 6
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12 | 10, 11 | mpan 424 |
. . . . 5
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13 | 3anass 984 |
. . . . 5
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14 | 12, 13 | bitrdi 196 |
. . . 4
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15 | 9, 14 | syl 14 |
. . 3
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16 | ibar 301 |
. . . 4
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17 | 16 | bicomd 141 |
. . 3
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18 | 15, 17 | sylan9bbr 463 |
. 2
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19 | 8, 18 | bitr4d 191 |
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 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 ax-pre-mulext 7976 ax-arch 7977 |
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 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-po 4321 df-iso 4322 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-1st 6180 df-2nd 6181 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-div 8678 df-inn 8969 df-n0 9227 df-z 9304 df-q 9671 df-rp 9706 df-fz 10061 df-fl 10325 df-mod 10380 |
This theorem is referenced by: zmodidfzo 10410 |
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