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Theorem subrngsubg 20552
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 20551 . . 3 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2 rnggrp 20155 . . 3 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2737 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrngss 20548 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2737 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrngrng 20550 . . 3 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Rng)
8 rnggrp 20155 . . 3 ((𝑅s 𝐴) ∈ Rng → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 19144 . 2 (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1344 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Grpcgrp 18951  SubGrpcsubg 19138  Rngcrng 20149  SubRngcsubrng 20545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-subg 19141  df-abl 19801  df-rng 20150  df-subrng 20546
This theorem is referenced by:  subrngringnsg  20553  subrngbas  20554  subrng0  20555  subrngacl  20556  issubrng2  20558  subrngint  20560  rhmimasubrng  20566  rng2idl0  21277  rng2idlsubg0  21280  rngqiprnglinlem2  21302  rngqiprng  21306  rng2idl1cntr  21315
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