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Mirrors > Home > MPE Home > Th. List > subrgsubg | Structured version Visualization version GIF version |
Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.) |
Ref | Expression |
---|---|
subrgsubg | β’ (π΄ β (SubRingβπ ) β π΄ β (SubGrpβπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgrcl 20328 | . . 3 β’ (π΄ β (SubRingβπ ) β π β Ring) | |
2 | ringgrp 20063 | . . 3 β’ (π β Ring β π β Grp) | |
3 | 1, 2 | syl 17 | . 2 β’ (π΄ β (SubRingβπ ) β π β Grp) |
4 | eqid 2732 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
5 | 4 | subrgss 20324 | . 2 β’ (π΄ β (SubRingβπ ) β π΄ β (Baseβπ )) |
6 | eqid 2732 | . . . 4 β’ (π βΎs π΄) = (π βΎs π΄) | |
7 | 6 | subrgring 20326 | . . 3 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Ring) |
8 | ringgrp 20063 | . . 3 β’ ((π βΎs π΄) β Ring β (π βΎs π΄) β Grp) | |
9 | 7, 8 | syl 17 | . 2 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Grp) |
10 | 4 | issubg 19008 | . 2 β’ (π΄ β (SubGrpβπ ) β (π β Grp β§ π΄ β (Baseβπ ) β§ (π βΎs π΄) β Grp)) |
11 | 3, 5, 9, 10 | syl3anbrc 1343 | 1 β’ (π΄ β (SubRingβπ ) β π΄ β (SubGrpβπ )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2106 β wss 3948 βcfv 6543 (class class class)co 7411 Basecbs 17146 βΎs cress 17175 Grpcgrp 18821 SubGrpcsubg 19002 Ringcrg 20058 SubRingcsubrg 20319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-subg 19005 df-ring 20060 df-subrg 20321 |
This theorem is referenced by: subrg0 20330 subrgbas 20332 subrgacl 20334 issubrg2 20343 subrgint 20346 resrhm 20352 resrhm2b 20353 rhmima 20355 subdrgint 20423 primefld0cl 20426 abvres 20451 zsssubrg 21009 gzrngunitlem 21016 zringlpirlem1 21038 zringcyg 21045 zringsubgval 21046 prmirred 21050 zndvds 21111 resubgval 21168 rzgrp 21182 issubassa2 21452 resspsrmul 21543 subrgpsr 21545 mplbas2 21603 gsumply1subr 21763 subrgnrg 24197 sranlm 24208 clmsub 24603 clmneg 24604 clmabs 24606 clmsubcl 24609 isncvsngp 24673 cphsqrtcl3 24711 tcphcph 24761 plypf1 25733 dvply2g 25805 taylply2 25887 circgrp 26068 circsubm 26069 jensenlem2 26499 amgmlem 26501 lgseisenlem4 26888 qrng0 27131 qrngneg 27133 subrgchr 32427 1fldgenq 32453 nn0archi 32503 idlinsubrg 32594 ressply10g 32701 ressply1invg 32703 ressply1sub 32704 drgext0gsca 32737 fedgmullem1 32773 fedgmullem2 32774 evls1fldgencl 32804 irngss 32811 algextdeglem1 32833 algextdeglem2 32834 algextdeglem3 32835 algextdeglem4 32836 algextdeglem5 32837 rezh 33020 qqhcn 33040 qqhucn 33041 fsumcnsrcl 41990 cnsrplycl 41991 rngunsnply 41997 amgmwlem 47927 |
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