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Theorem subrgsubg 13286
Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
Assertion
Ref Expression
subrgsubg  |-  ( A  e.  (SubRing `  R
)  ->  A  e.  (SubGrp `  R ) )

Proof of Theorem subrgsubg
StepHypRef Expression
1 subrgrcl 13285 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2 ringgrp 13115 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 14 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Grp )
4 eqid 2177 . . 3  |-  ( Base `  R )  =  (
Base `  R )
54subrgss 13281 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
6 eqid 2177 . . . 4  |-  ( Rs  A )  =  ( Rs  A )
76subrgring 13283 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Ring )
8 ringgrp 13115 . . 3  |-  ( ( Rs  A )  e.  Ring  -> 
( Rs  A )  e.  Grp )
97, 8syl 14 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Grp )
104issubg 12964 . 2  |-  ( A  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  A  C_  ( Base `  R )  /\  ( Rs  A )  e.  Grp ) )
113, 5, 9, 10syl3anbrc 1181 1  |-  ( A  e.  (SubRing `  R
)  ->  A  e.  (SubGrp `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2148    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s 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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