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Mirrors > Home > ILE Home > Th. List > subrgintm | Unicode version |
Description: The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
subrgintm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgsubg 13308 |
. . . . 5
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2 | 1 | ssriv 3159 |
. . . 4
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3 | sstr 3163 |
. . . 4
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4 | 2, 3 | mpan2 425 |
. . 3
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5 | subgintm 13011 |
. . 3
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6 | 4, 5 | sylan 283 |
. 2
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7 | ssel2 3150 |
. . . . . 6
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8 | 7 | adantlr 477 |
. . . . 5
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9 | eqid 2177 |
. . . . . 6
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10 | 9 | subrg1cl 13310 |
. . . . 5
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11 | 8, 10 | syl 14 |
. . . 4
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12 | 11 | ralrimiva 2550 |
. . 3
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13 | ssel 3149 |
. . . . . . 7
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14 | subrgrcl 13307 |
. . . . . . 7
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15 | 13, 14 | syl6 33 |
. . . . . 6
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16 | 15 | exlimdv 1819 |
. . . . 5
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17 | 16 | imp 124 |
. . . 4
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18 | ringsrg 13177 |
. . . 4
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19 | eqid 2177 |
. . . . . 6
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20 | 19, 9 | srgidcl 13112 |
. . . . 5
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21 | elintg 3852 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 17, 18, 22 | 3syl 17 |
. . 3
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24 | 12, 23 | mpbird 167 |
. 2
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25 | 8 | adantlr 477 |
. . . . . 6
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26 | simprl 529 |
. . . . . . 7
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27 | elinti 3853 |
. . . . . . . 8
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28 | 27 | imp 124 |
. . . . . . 7
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29 | 26, 28 | sylan 283 |
. . . . . 6
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30 | simprr 531 |
. . . . . . 7
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31 | elinti 3853 |
. . . . . . . 8
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32 | 31 | imp 124 |
. . . . . . 7
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33 | 30, 32 | sylan 283 |
. . . . . 6
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34 | eqid 2177 |
. . . . . . 7
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35 | 34 | subrgmcl 13314 |
. . . . . 6
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36 | 25, 29, 33, 35 | syl3anc 1238 |
. . . . 5
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37 | 36 | ralrimiva 2550 |
. . . 4
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38 | simplr 528 |
. . . . . 6
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39 | eleq1w 2238 |
. . . . . . . 8
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40 | 39 | cbvexv 1918 |
. . . . . . 7
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41 | 36 | elexd 2750 |
. . . . . . . . 9
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42 | 41 | ex 115 |
. . . . . . . 8
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43 | 42 | exlimdv 1819 |
. . . . . . 7
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44 | 40, 43 | biimtrrid 153 |
. . . . . 6
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45 | 38, 44 | mpd 13 |
. . . . 5
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46 | elintg 3852 |
. . . . 5
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47 | 45, 46 | syl 14 |
. . . 4
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48 | 37, 47 | mpbird 167 |
. . 3
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49 | 48 | ralrimivva 2559 |
. 2
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50 | 19, 9, 34 | issubrg2 13322 |
. . 3
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51 | 17, 50 | syl 14 |
. 2
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52 | 6, 24, 49, 51 | mpbir3and 1180 |
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 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
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 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-iress 12464 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-subg 12983 df-cmn 13043 df-abl 13044 df-mgp 13084 df-ur 13096 df-srg 13100 df-ring 13134 df-subrg 13300 |
This theorem is referenced by: subrgin 13325 |
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