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Theorem subrngsubg 20448
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 20447 . . 3 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
2 rnggrp 20059 . . 3 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
31, 2syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Grp)
4 eqid 2731 . . 3 (Base‘𝑅) = (Base‘𝑅)
54subrngss 20444 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ (Base‘𝑅))
6 eqid 2731 . . . 4 (𝑅s 𝐴) = (𝑅s 𝐴)
76subrngrng 20446 . . 3 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Rng)
8 rnggrp 20059 . . 3 ((𝑅s 𝐴) ∈ Rng → (𝑅s 𝐴) ∈ Grp)
97, 8syl 17 . 2 (𝐴 ∈ (SubRng‘𝑅) → (𝑅s 𝐴) ∈ Grp)
104issubg 19049 . 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 2105  wss 3948  cfv 6543  (class class class)co 7412  Basecbs 17151  s cress 17180  Grpcgrp 18861  SubGrpcsubg 19043  Rngcrng 20053  SubRngcsubrng 20441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-subg 19046  df-abl 19699  df-rng 20054  df-subrng 20442
This theorem is referenced by:  subrngringnsg  20449  subrngbas  20450  subrng0  20451  subrngacl  20452  issubrng2  20454  subrngint  20456  rhmimasubrng  20462  rng2idl0  21120  rng2idlsubg0  21123  rngqiprnglinlem2  21141  rngqiprng  21145  rng2idl1cntr  21154
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