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Theorem subrngsubg 20569
Description: A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
Assertion
Ref Expression
subrngsubg (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))

Proof of Theorem subrngsubg
StepHypRef Expression
1 subrngrcl 20568 . . 3 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2 rnggrp 20176 . . 3 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2735 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrngss 20565 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2735 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrngrng 20567 . . 3 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Rng)
8 rnggrp 20176 . . 3 ((𝑅s 𝐴) ∈ Rng → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 19157 . 2 (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1342 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  Grpcgrp 18964  SubGrpcsubg 19151  Rngcrng 20170  SubRngcsubrng 20562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-subg 19154  df-abl 19816  df-rng 20171  df-subrng 20563
This theorem is referenced by:  subrngringnsg  20570  subrngbas  20571  subrng0  20572  subrngacl  20573  issubrng2  20575  subrngint  20577  rhmimasubrng  20583  rng2idl0  21295  rng2idlsubg0  21298  rngqiprnglinlem2  21320  rngqiprng  21324  rng2idl1cntr  21333
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