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Theorem subrgsubg 15829
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 15828 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2 rnggrp 15624 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Grp )
4 eqid 2404 . . 3  |-  ( Base `  R )  =  (
Base `  R )
54subrgss 15824 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
6 eqid 2404 . . . 4  |-  ( Rs  A )  =  ( Rs  A )
76subrgrng 15826 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Ring )
8 rnggrp 15624 . . 3  |-  ( ( Rs  A )  e.  Ring  -> 
( Rs  A )  e.  Grp )
97, 8syl 16 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Grp )
104issubg 14899 . 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 1721    C_ wss 3280   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   Grpcgrp 14640  SubGrpcsubg 14893   Ringcrg 15615  SubRingcsubrg 15819
This theorem is referenced by:  subrg0  15830  subrgbas  15832  subrgacl  15834  issubrg2  15843  subrgint  15845  resrhm  15852  rhmima  15854  abvres  15882  issubassa2  16358  resspsrmul  16435  subrgpsr  16437  mplbas2  16486  zsssubrg  16712  gzrngunitlem  16718  zlpirlem1  16723  zcyg  16727  prmirred  16730  expghm  16732  mulgrhm2  16743  zndvds  16785  frgpcyg  16809  subrgnrg  18662  sranlm  18673  clmsub  19058  clmneg  19059  clmabs  19060  clmsubcl  19063  cphsqrcl3  19103  tchcph  19147  plypf1  20084  dvply2g  20155  taylply2  20237  jensenlem2  20779  amgmlem  20781  lgseisenlem4  21089  dchrisum0flblem1  21155  qrng0  21268  qrngneg  21270  subrgchr  24183  rezh  24308  qqhcn  24328  qqhucn  24329  fsumcnsrcl  27239  cnsrplycl  27240  rngunsnply  27246
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-subg 14896  df-rng 15618  df-subrg 15821
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