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Mirrors > Home > ILE Home > Th. List > dvrvald | Unicode version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrvald.b |
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dvrvald.t |
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dvrvald.u |
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dvrvald.i |
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dvrvald.d |
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dvrvald.r |
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dvrvald.x |
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dvrvald.y |
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Ref | Expression |
---|---|
dvrvald |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrvald.b |
. . 3
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2 | dvrvald.t |
. . 3
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3 | dvrvald.u |
. . 3
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4 | dvrvald.i |
. . 3
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5 | dvrvald.d |
. . 3
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6 | dvrvald.r |
. . . 4
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7 | ringsrg 13360 |
. . . 4
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8 | 6, 7 | syl 14 |
. . 3
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9 | 1, 2, 3, 4, 5, 8 | dvrfvald 13444 |
. 2
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10 | simpl 109 |
. . . 4
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11 | fveq2 5530 |
. . . . 5
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12 | 11 | adantl 277 |
. . . 4
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13 | 10, 12 | oveq12d 5909 |
. . 3
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14 | 13 | adantl 277 |
. 2
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15 | dvrvald.x |
. 2
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16 | dvrvald.y |
. 2
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17 | 2 | oveqd 5908 |
. . 3
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18 | 15, 1 | eleqtrd 2268 |
. . . 4
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19 | eqidd 2190 |
. . . . 5
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20 | 16, 3 | eleqtrd 2268 |
. . . . . . 7
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21 | eqid 2189 |
. . . . . . . 8
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22 | eqid 2189 |
. . . . . . . 8
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23 | 21, 22 | unitinvcl 13434 |
. . . . . . 7
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24 | 6, 20, 23 | syl2anc 411 |
. . . . . 6
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25 | 4 | fveq1d 5532 |
. . . . . 6
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26 | 24, 25, 3 | 3eltr4d 2273 |
. . . . 5
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27 | 19, 3, 8, 26 | unitcld 13419 |
. . . 4
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28 | eqid 2189 |
. . . . 5
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29 | eqid 2189 |
. . . . 5
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30 | 28, 29 | ringcl 13328 |
. . . 4
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31 | 6, 18, 27, 30 | syl3anc 1249 |
. . 3
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32 | 17, 31 | eqeltrd 2266 |
. 2
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33 | 9, 14, 15, 16, 32 | ovmpod 6019 |
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-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-addcom 7929 ax-addass 7931 ax-i2m1 7934 ax-0lt1 7935 ax-0id 7937 ax-rnegex 7938 ax-pre-ltirr 7941 ax-pre-lttrn 7943 ax-pre-ltadd 7945 |
This theorem depends on definitions: df-bi 117 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-nul 3438 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-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 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-tpos 6264 df-pnf 8012 df-mnf 8013 df-ltxr 8015 df-inn 8938 df-2 8996 df-3 8997 df-ndx 12483 df-slot 12484 df-base 12486 df-sets 12487 df-iress 12488 df-plusg 12568 df-mulr 12569 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-minusg 12915 df-cmn 13186 df-abl 13187 df-mgp 13236 df-ur 13275 df-srg 13279 df-ring 13313 df-oppr 13379 df-dvdsr 13400 df-unit 13401 df-invr 13432 df-dvr 13443 |
This theorem is referenced by: dvrcl 13446 unitdvcl 13447 dvrid 13448 dvr1 13449 dvrass 13450 dvrcan1 13451 dvrdir 13454 rdivmuldivd 13455 ringinvdv 13456 subrgdv 13546 |
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