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Theorem subrgsubg 20329
Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
Assertion
Ref Expression
subrgsubg (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 ∈ (SubGrpβ€˜π‘…))

Proof of Theorem subrgsubg
StepHypRef Expression
1 subrgrcl 20328 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Ring)
2 ringgrp 20063 . . 3 (𝑅 ∈ Ring β†’ 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑅 ∈ Grp)
4 eqid 2732 . . 3 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
54subrgss 20324 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 βŠ† (Baseβ€˜π‘…))
6 eqid 2732 . . . 4 (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs 𝐴)
76subrgring 20326 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐴) ∈ Ring)
8 ringgrp 20063 . . 3 ((𝑅 β†Ύs 𝐴) ∈ Ring β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐴) ∈ Grp)
104issubg 19008 . 2 (𝐴 ∈ (SubGrpβ€˜π‘…) ↔ (𝑅 ∈ Grp ∧ 𝐴 βŠ† (Baseβ€˜π‘…) ∧ (𝑅 β†Ύs 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1343 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝐴 ∈ (SubGrpβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2106   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146   β†Ύs cress 17175  Grpcgrp 18821  SubGrpcsubg 19002  Ringcrg 20058  SubRingcsubrg 20319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-subg 19005  df-ring 20060  df-subrg 20321
This theorem is referenced by:  subrg0  20330  subrgbas  20332  subrgacl  20334  issubrg2  20343  subrgint  20346  resrhm  20352  resrhm2b  20353  rhmima  20355  subdrgint  20423  primefld0cl  20426  abvres  20451  zsssubrg  21009  gzrngunitlem  21016  zringlpirlem1  21038  zringcyg  21045  zringsubgval  21046  prmirred  21050  zndvds  21111  resubgval  21168  rzgrp  21182  issubassa2  21452  resspsrmul  21543  subrgpsr  21545  mplbas2  21603  gsumply1subr  21763  subrgnrg  24197  sranlm  24208  clmsub  24603  clmneg  24604  clmabs  24606  clmsubcl  24609  isncvsngp  24673  cphsqrtcl3  24711  tcphcph  24761  plypf1  25733  dvply2g  25805  taylply2  25887  circgrp  26068  circsubm  26069  jensenlem2  26499  amgmlem  26501  lgseisenlem4  26888  qrng0  27131  qrngneg  27133  subrgchr  32427  1fldgenq  32453  nn0archi  32503  idlinsubrg  32594  ressply10g  32701  ressply1invg  32703  ressply1sub  32704  drgext0gsca  32737  fedgmullem1  32773  fedgmullem2  32774  evls1fldgencl  32804  irngss  32811  algextdeglem1  32833  algextdeglem2  32834  algextdeglem3  32835  algextdeglem4  32836  algextdeglem5  32837  rezh  33020  qqhcn  33040  qqhucn  33041  fsumcnsrcl  41990  cnsrplycl  41991  rngunsnply  41997  amgmwlem  47927
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