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Theorem subrngringnsg 20554
Description: A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
Assertion
Ref Expression
subrngringnsg (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))

Proof of Theorem subrngringnsg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrngsubg 20553 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 subrngrcl 20552 . . . . . . . . 9 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3 rngabl 20153 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
42, 3syl 17 . . . . . . . 8 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel)
543anim1i 1152 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
653expb 1120 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
7 eqid 2736 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2736 . . . . . . 7 (+g𝑅) = (+g𝑅)
97, 8ablcom 19818 . . . . . 6 ((𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)𝑦) = (𝑦(+g𝑅)𝑥))
106, 9syl 17 . . . . 5 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g𝑅)𝑦) = (𝑦(+g𝑅)𝑥))
1110eleq1d 2825 . . . 4 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g𝑅)𝑦) ∈ 𝐴 ↔ (𝑦(+g𝑅)𝑥) ∈ 𝐴))
1211biimpd 229 . . 3 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(+g𝑅)𝑦) ∈ 𝐴 → (𝑦(+g𝑅)𝑥) ∈ 𝐴))
1312ralrimivva 3201 . 2 (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) ∈ 𝐴 → (𝑦(+g𝑅)𝑥) ∈ 𝐴))
147, 8isnsg2 19175 . 2 (𝐴 ∈ (NrmSGrp‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) ∈ 𝐴 → (𝑦(+g𝑅)𝑥) ∈ 𝐴)))
151, 13, 14sylanbrc 583 1 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  SubGrpcsubg 19139  NrmSGrpcnsg 19140  Abelcabl 19800  Rngcrng 20150  SubRngcsubrng 20546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-subg 19142  df-nsg 19143  df-cmn 19801  df-abl 19802  df-rng 20151  df-subrng 20547
This theorem is referenced by:  rng2idlnsg  21277  rng2idlsubgnsg  21280
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