MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrgsubg Unicode version

Theorem subrgsubg 15801
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 15800 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2 rnggrp 15596 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Grp )
4 eqid 2387 . . 3  |-  ( Base `  R )  =  (
Base `  R )
54subrgss 15796 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
6 eqid 2387 . . . 4  |-  ( Rs  A )  =  ( Rs  A )
76subrgrng 15798 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Ring )
8 rnggrp 15596 . . 3  |-  ( ( Rs  A )  e.  Ring  -> 
( Rs  A )  e.  Grp )
97, 8syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Grp )
104issubg 14871 . 2  |-  ( A  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  A  C_  ( Base `  R )  /\  ( Rs  A )  e.  Grp ) )
113, 5, 9, 10syl3anbrc 1138 1  |-  ( A  e.  (SubRing `  R
)  ->  A  e.  (SubGrp `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   ↾s cress 13397   Grpcgrp 14612  SubGrpcsubg 14865   Ringcrg 15587  SubRingcsubrg 15791
This theorem is referenced by:  subrg0  15802  subrgbas  15804  subrgacl  15806  issubrg2  15815  subrgint  15817  resrhm  15824  rhmima  15826  abvres  15854  issubassa2  16330  resspsrmul  16407  subrgpsr  16409  mplbas2  16458  zsssubrg  16680  gzrngunitlem  16686  zlpirlem1  16691  zcyg  16695  prmirred  16698  expghm  16700  mulgrhm2  16711  zndvds  16753  frgpcyg  16777  subrgnrg  18580  sranlm  18591  clmsub  18976  clmneg  18977  clmabs  18978  clmsubcl  18981  cphsqrcl3  19021  tchcph  19065  plypf1  19998  dvply2g  20069  taylply2  20151  jensenlem2  20693  amgmlem  20695  lgseisenlem4  21003  dchrisum0flblem1  21069  qrng0  21182  qrngneg  21184  qqhcn  24174  qqhucn  24175  rrhre  24183  fsumcnsrcl  27040  cnsrplycl  27041  rngunsnply  27047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-subg 14868  df-rng 15590  df-subrg 15793
  Copyright terms: Public domain W3C validator