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Theorem subrngsubg 20493
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 20492 . . 3 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2 rnggrp 20102 . . 3 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2725 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrngss 20489 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2725 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrngrng 20491 . . 3 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Rng)
8 rnggrp 20102 . . 3 ((𝑅s 𝐴) ∈ Rng → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 19085 . 2 (𝐴 ∈ (SubGrp‘𝑅) ↔ (𝑅 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝑅) ∧ (𝑅s 𝐴) ∈ Grp))
113, 5, 9, 10syl3anbrc 1340 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3939  cfv 6543  (class class class)co 7416  Basecbs 17179  s cress 17208  Grpcgrp 18894  SubGrpcsubg 19079  Rngcrng 20096  SubRngcsubrng 20486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-subg 19082  df-abl 19742  df-rng 20097  df-subrng 20487
This theorem is referenced by:  subrngringnsg  20494  subrngbas  20495  subrng0  20496  subrngacl  20497  issubrng2  20499  subrngint  20501  rhmimasubrng  20507  rng2idl0  21165  rng2idlsubg0  21168  rngqiprnglinlem2  21186  rngqiprng  21190  rng2idl1cntr  21199
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