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Theorem subrgsubg 15551
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 15550 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
2 rnggrp 15346 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
31, 2syl 15 . 2  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Grp )
4 eqid 2283 . . 3  |-  ( Base `  R )  =  (
Base `  R )
54subrgss 15546 . 2  |-  ( A  e.  (SubRing `  R
)  ->  A  C_  ( Base `  R ) )
6 eqid 2283 . . . 4  |-  ( Rs  A )  =  ( Rs  A )
76subrgrng 15548 . . 3  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Ring )
8 rnggrp 15346 . . 3  |-  ( ( Rs  A )  e.  Ring  -> 
( Rs  A )  e.  Grp )
97, 8syl 15 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( Rs  A
)  e.  Grp )
104issubg 14621 . 2  |-  ( A  e.  (SubGrp `  R
)  <->  ( R  e. 
Grp  /\  A  C_  ( Base `  R )  /\  ( Rs  A )  e.  Grp ) )
113, 5, 9, 10syl3anbrc 1136 1  |-  ( A  e.  (SubRing `  R
)  ->  A  e.  (SubGrp `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615   Ringcrg 15337  SubRingcsubrg 15541
This theorem is referenced by:  subrg0  15552  subrgbas  15554  subrgacl  15556  issubrg2  15565  subrgint  15567  resrhm  15574  rhmima  15576  abvres  15604  issubassa2  16084  resspsrmul  16161  subrgpsr  16163  mplbas2  16212  zsssubrg  16430  gzrngunitlem  16436  zlpirlem1  16441  zcyg  16445  prmirred  16448  expghm  16450  mulgrhm2  16461  zndvds  16503  frgpcyg  16527  subrgnrg  18184  sranlm  18195  clmsub  18578  clmneg  18579  clmabs  18580  clmsubcl  18583  cphsqrcl3  18623  tchcph  18667  plypf1  19594  dvply2g  19665  taylply2  19747  jensenlem2  20282  amgmlem  20284  lgseisenlem4  20591  dchrisum0flblem1  20657  qrng0  20770  qrngneg  20772  fsumcnsrcl  27371  cnsrplycl  27372  rngunsnply  27378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-subg 14618  df-rng 15340  df-subrg 15543
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