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Theorem subrgsubg 19470
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 19469 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
2 ringgrp 19231 . . 3 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2818 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrgss 19465 . 2 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2818 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrgring 19467 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Ring)
8 ringgrp 19231 . . 3 ((𝑅s 𝐴) ∈ Ring → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 18217 . 2 (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1335 1 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  s cress 16472  Grpcgrp 18041  SubGrpcsubg 18211  Ringcrg 19226  SubRingcsubrg 19460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-subg 18214  df-ring 19228  df-subrg 19462
This theorem is referenced by:  subrg0  19471  subrgbas  19473  subrgacl  19475  issubrg2  19484  subrgint  19486  resrhm  19493  rhmima  19495  subdrgint  19511  primefld0cl  19514  abvres  19539  issubassa2  20049  resspsrmul  20125  subrgpsr  20127  mplbas2  20179  gsumply1subr  20330  zsssubrg  20531  gzrngunitlem  20538  zringlpirlem1  20559  zringcyg  20566  prmirred  20570  zndvds  20624  resubgval  20681  rzgrp  20695  subrgnrg  23209  sranlm  23220  clmsub  23611  clmneg  23612  clmabs  23614  clmsubcl  23617  isncvsngp  23680  cphsqrtcl3  23718  tcphcph  23767  plypf1  24729  dvply2g  24801  taylply2  24883  circgrp  25063  circsubm  25064  jensenlem2  25492  amgmlem  25494  lgseisenlem4  25881  qrng0  26124  qrngneg  26126  subrgchr  30792  nn0archi  30843  drgext0gsca  30893  fedgmullem1  30924  fedgmullem2  30925  rezh  31111  qqhcn  31131  qqhucn  31132  selvval2lem4  39014  fsumcnsrcl  39644  cnsrplycl  39645  rngunsnply  39651  zringsubgval  44377  amgmwlem  44831
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