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Mirrors > Home > ILE Home > Th. List > ringidss | Unicode version |
Description: A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
ringidss.g |
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ringidss.b |
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ringidss.u |
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Ref | Expression |
---|---|
ringidss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 |
. 2
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2 | eqid 2177 |
. 2
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3 | eqid 2177 |
. 2
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4 | simp3 999 |
. . 3
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5 | ringidss.g |
. . . . 5
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6 | 5 | a1i 9 |
. . . 4
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7 | eqid 2177 |
. . . . . 6
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8 | ringidss.b |
. . . . . 6
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9 | 7, 8 | mgpbasg 13136 |
. . . . 5
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10 | 9 | 3ad2ant1 1018 |
. . . 4
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11 | 7 | mgpex 13135 |
. . . . 5
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12 | 11 | 3ad2ant1 1018 |
. . . 4
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13 | simp2 998 |
. . . 4
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14 | 6, 10, 12, 13 | ressbas2d 12528 |
. . 3
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15 | 4, 14 | eleqtrd 2256 |
. 2
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16 | 14, 13 | eqsstrrd 3193 |
. . . 4
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17 | 16 | sselda 3156 |
. . 3
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18 | eqid 2177 |
. . . . . . . . 9
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19 | 7, 18 | mgpplusgg 13134 |
. . . . . . . 8
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20 | 19 | 3ad2ant1 1018 |
. . . . . . 7
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21 | basfn 12520 |
. . . . . . . . . 10
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22 | simp1 997 |
. . . . . . . . . . 11
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23 | 22 | elexd 2751 |
. . . . . . . . . 10
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24 | funfvex 5533 |
. . . . . . . . . . 11
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25 | 24 | funfni 5317 |
. . . . . . . . . 10
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26 | 21, 23, 25 | sylancr 414 |
. . . . . . . . 9
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27 | 8, 26 | eqeltrid 2264 |
. . . . . . . 8
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28 | 27, 13 | ssexd 4144 |
. . . . . . 7
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29 | 6, 20, 28, 12 | ressplusgd 12587 |
. . . . . 6
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30 | 29 | adantr 276 |
. . . . 5
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31 | 30 | oveqd 5892 |
. . . 4
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32 | ringidss.u |
. . . . . 6
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33 | 8, 18, 32 | ringlidm 13206 |
. . . . 5
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34 | 33 | 3ad2antl1 1159 |
. . . 4
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35 | 31, 34 | eqtr3d 2212 |
. . 3
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36 | 17, 35 | syldan 282 |
. 2
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37 | 30 | oveqd 5892 |
. . . 4
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38 | 8, 18, 32 | ringridm 13207 |
. . . . 5
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39 | 38 | 3ad2antl1 1159 |
. . . 4
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40 | 37, 39 | eqtr3d 2212 |
. . 3
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41 | 17, 40 | syldan 282 |
. 2
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42 | 1, 2, 3, 15, 36, 41 | ismgmid2 12799 |
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-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-pre-ltirr 7923 ax-pre-ltadd 7927 |
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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-inn 8920 df-2 8978 df-3 8979 df-ndx 12465 df-slot 12466 df-base 12468 df-sets 12469 df-iress 12470 df-plusg 12549 df-mulr 12550 df-0g 12707 df-mgm 12775 df-sgrp 12808 df-mnd 12818 df-mgp 13131 df-ur 13143 df-ring 13181 |
This theorem is referenced by: unitgrpid 13287 |
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