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Mirrors > Home > ILE Home > Th. List > modmulconst | Unicode version |
Description: Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
Ref | Expression |
---|---|
modmulconst |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnz 9297 |
. . . . 5
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2 | 1 | adantl 277 |
. . . 4
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3 | zsubcl 9319 |
. . . . . 6
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4 | 3 | 3adant3 1019 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | adantr 276 |
. . . 4
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6 | nnz 9297 |
. . . . . . 7
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7 | nnne0 8972 |
. . . . . . 7
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8 | 6, 7 | jca 306 |
. . . . . 6
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9 | 8 | 3ad2ant3 1022 |
. . . . 5
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10 | 9 | adantr 276 |
. . . 4
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11 | dvdscmulr 11854 |
. . . . 5
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12 | 11 | bicomd 141 |
. . . 4
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13 | 2, 5, 10, 12 | syl3anc 1249 |
. . 3
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14 | zcn 9283 |
. . . . . . . 8
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15 | zcn 9283 |
. . . . . . . 8
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16 | nncn 8952 |
. . . . . . . 8
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17 | 14, 15, 16 | 3anim123i 1186 |
. . . . . . 7
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18 | 3anrot 985 |
. . . . . . 7
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19 | 17, 18 | sylibr 134 |
. . . . . 6
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20 | subdi 8367 |
. . . . . 6
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21 | 19, 20 | syl 14 |
. . . . 5
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22 | 21 | adantr 276 |
. . . 4
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23 | 22 | breq2d 4030 |
. . 3
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24 | 13, 23 | bitrd 188 |
. 2
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25 | simpr 110 |
. . 3
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26 | simp1 999 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 26 | adantr 276 |
. . 3
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28 | simp2 1000 |
. . . 4
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29 | 28 | adantr 276 |
. . 3
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30 | moddvds 11833 |
. . 3
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31 | 25, 27, 29, 30 | syl3anc 1249 |
. 2
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32 | simpl3 1004 |
. . . 4
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33 | 32, 25 | nnmulcld 8993 |
. . 3
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34 | 6 | 3ad2ant3 1022 |
. . . . 5
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35 | 34, 26 | zmulcld 9406 |
. . . 4
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36 | 35 | adantr 276 |
. . 3
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37 | 34, 28 | zmulcld 9406 |
. . . 4
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38 | 37 | adantr 276 |
. . 3
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39 | moddvds 11833 |
. . 3
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40 | 33, 36, 38, 39 | syl3anc 1249 |
. 2
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41 | 24, 31, 40 | 3bitr4d 220 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 ax-arch 7955 |
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 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 df-n0 9202 df-z 9279 df-q 9645 df-rp 9679 df-fl 10296 df-mod 10349 df-dvds 11822 |
This theorem is referenced by: (None) |
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