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Mirrors > Home > ILE Home > Th. List > subrgsubg | 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 13285 | . . 3 β’ (π΄ β (SubRingβπ ) β π β Ring) | |
2 | ringgrp 13115 | . . 3 β’ (π β Ring β π β Grp) | |
3 | 1, 2 | syl 14 | . 2 β’ (π΄ β (SubRingβπ ) β π β Grp) |
4 | eqid 2177 | . . 3 β’ (Baseβπ ) = (Baseβπ ) | |
5 | 4 | subrgss 13281 | . 2 β’ (π΄ β (SubRingβπ ) β π΄ β (Baseβπ )) |
6 | eqid 2177 | . . . 4 β’ (π βΎs π΄) = (π βΎs π΄) | |
7 | 6 | subrgring 13283 | . . 3 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Ring) |
8 | ringgrp 13115 | . . 3 β’ ((π βΎs π΄) β Ring β (π βΎs π΄) β Grp) | |
9 | 7, 8 | syl 14 | . 2 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Grp) |
10 | 4 | issubg 12964 | . 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 1181 | 1 β’ (π΄ β (SubRingβπ ) β π΄ β (SubGrpβπ )) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β wcel 2148 β wss 3129 βcfv 5215 (class class class)co 5872 Basecbs 12454 βΎs cress 12455 Grpcgrp 12809 SubGrpcsubg 12958 Ringcrg 13110 SubRingcsubrg 13276 |
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-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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 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-fv 5223 df-ov 5875 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-mulr 12542 df-subg 12961 df-ring 13112 df-subrg 13278 |
This theorem is referenced by: subrg0 13287 subrgbas 13289 subrgacl 13291 issubrg2 13300 subrgintm 13302 zsssubrg 13348 zringsubgval 13364 |
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