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Mirrors > Home > ILE Home > Th. List > subrgdv | Unicode version |
Description: A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
subrgdv.1 |
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subrgdv.2 |
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subrgdv.3 |
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subrgdv.4 |
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Ref | Expression |
---|---|
subrgdv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgdv.1 |
. . . . . 6
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2 | eqid 2177 |
. . . . . 6
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3 | subrgdv.3 |
. . . . . 6
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4 | eqid 2177 |
. . . . . 6
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5 | 1, 2, 3, 4 | subrginv 13296 |
. . . . 5
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6 | 5 | 3adant2 1016 |
. . . 4
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7 | 6 | oveq2d 5888 |
. . 3
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8 | subrgrcl 13285 |
. . . . . 6
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9 | eqid 2177 |
. . . . . . 7
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10 | 1, 9 | ressmulrg 12595 |
. . . . . 6
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11 | 8, 10 | mpdan 421 |
. . . . 5
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12 | 11 | 3ad2ant1 1018 |
. . . 4
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13 | 12 | oveqd 5889 |
. . 3
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14 | 7, 13 | eqtrd 2210 |
. 2
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15 | eqidd 2178 |
. . 3
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16 | eqidd 2178 |
. . 3
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17 | eqidd 2178 |
. . 3
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18 | eqidd 2178 |
. . 3
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19 | subrgdv.2 |
. . . 4
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20 | 19 | a1i 9 |
. . 3
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21 | 8 | 3ad2ant1 1018 |
. . 3
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22 | eqid 2177 |
. . . . . 6
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23 | 22 | subrgss 13281 |
. . . . 5
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24 | 23 | 3ad2ant1 1018 |
. . . 4
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25 | simp2 998 |
. . . 4
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26 | 24, 25 | sseldd 3156 |
. . 3
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27 | eqid 2177 |
. . . . . 6
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28 | 1, 27, 3 | subrguss 13295 |
. . . . 5
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29 | 28 | 3ad2ant1 1018 |
. . . 4
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30 | simp3 999 |
. . . 4
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31 | 29, 30 | sseldd 3156 |
. . 3
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32 | 15, 16, 17, 18, 20, 21, 26, 31 | dvrvald 13234 |
. 2
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33 | eqidd 2178 |
. . 3
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34 | eqidd 2178 |
. . 3
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35 | 3 | a1i 9 |
. . 3
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36 | eqidd 2178 |
. . 3
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37 | subrgdv.4 |
. . . 4
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38 | 37 | a1i 9 |
. . 3
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39 | 1 | subrgring 13283 |
. . . 4
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40 | 39 | 3ad2ant1 1018 |
. . 3
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41 | 1 | subrgbas 13289 |
. . . . 5
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42 | 41 | 3ad2ant1 1018 |
. . . 4
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43 | 25, 42 | eleqtrd 2256 |
. . 3
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44 | 33, 34, 35, 36, 38, 40, 43, 30 | dvrvald 13234 |
. 2
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45 | 14, 32, 44 | 3eqtr4d 2220 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-tpos 6243 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-subg 12961 df-cmn 13021 df-abl 13022 df-mgp 13062 df-ur 13074 df-srg 13078 df-ring 13112 df-oppr 13171 df-dvdsr 13189 df-unit 13190 df-invr 13221 df-dvr 13232 df-subrg 13278 |
This theorem is referenced by: (None) |
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