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Theorem subrgsubg 18710
Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
Assertion
Ref Expression
subrgsubg (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Proof of Theorem subrgsubg
StepHypRef Expression
1 subrgrcl 18709 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2 ringgrp 18476 . . 3 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2621 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrgss 18705 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2621 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrgring 18707 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Ring)
8 ringgrp 18476 . . 3 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 17518 . 2 (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1244 1 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wss 3556  cfv 5849  (class class class)co 6607  Basecbs 15784  s cress 15785  Grpcgrp 17346  SubGrpcsubg 17512  Ringcrg 18471  SubRingcsubrg 18700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fv 5857  df-ov 6610  df-subg 17515  df-ring 18473  df-subrg 18702
This theorem is referenced by:  subrg0  18711  subrgbas  18713  subrgacl  18715  issubrg2  18724  subrgint  18726  resrhm  18733  rhmima  18735  abvres  18763  issubassa2  19267  resspsrmul  19339  subrgpsr  19341  mplbas2  19392  gsumply1subr  19526  zsssubrg  19726  gzrngunitlem  19733  zringlpirlem1  19754  zringcyg  19761  prmirred  19765  zndvds  19820  resubgval  19877  subrgnrg  22390  sranlm  22401  clmsub  22793  clmneg  22794  clmabs  22796  clmsubcl  22799  isncvsngp  22862  cphsqrtcl3  22900  tchcph  22949  plypf1  23879  dvply2g  23951  taylply2  24033  circgrp  24209  circsubm  24210  rzgrp  24211  jensenlem2  24621  amgmlem  24623  lgseisenlem4  25010  qrng0  25217  qrngneg  25219  subrgchr  29591  nn0archi  29640  rezh  29809  qqhcn  29829  qqhucn  29830  fsumcnsrcl  37238  cnsrplycl  37239  rngunsnply  37245  zringsubgval  41487  amgmwlem  41867
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