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

Theorem subrgsubg 15866
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 15865 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2 rnggrp 15661 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Grp )
4 eqid 2435 . . 3  |-  ( Base `  R )  =  (
Base `  R )
54subrgss 15861 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
6 eqid 2435 . . . 4  |-  ( Rs  A )  =  ( Rs  A )
76subrgrng 15863 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Ring )
8 rnggrp 15661 . . 3  |-  ( ( Rs  A )  e.  Ring  -> 
( Rs  A )  e.  Grp )
97, 8syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Grp )
104issubg 14936 . 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 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Grpcgrp 14677  SubGrpcsubg 14930   Ringcrg 15652  SubRingcsubrg 15856
This theorem is referenced by:  subrg0  15867  subrgbas  15869  subrgacl  15871  issubrg2  15880  subrgint  15882  resrhm  15889  rhmima  15891  abvres  15919  issubassa2  16395  resspsrmul  16472  subrgpsr  16474  mplbas2  16523  zsssubrg  16749  gzrngunitlem  16755  zlpirlem1  16760  zcyg  16764  prmirred  16767  expghm  16769  mulgrhm2  16780  zndvds  16822  frgpcyg  16846  subrgnrg  18701  sranlm  18712  clmsub  19097  clmneg  19098  clmabs  19099  clmsubcl  19102  cphsqrcl3  19142  tchcph  19186  plypf1  20123  dvply2g  20194  taylply2  20276  jensenlem2  20818  amgmlem  20820  lgseisenlem4  21128  dchrisum0flblem1  21194  qrng0  21307  qrngneg  21309  subrgchr  24222  rezh  24347  qqhcn  24367  qqhucn  24368  fsumcnsrcl  27339  cnsrplycl  27340  rngunsnply  27346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-subg 14933  df-rng 15655  df-subrg 15858
  Copyright terms: Public domain W3C validator