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Theorem subrngringnsg 20570
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 20569 . 2 (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 subrngrcl 20568 . . . . . . . . 9 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
3 rngabl 20173 . . . . . . . . 9 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
42, 3syl 17 . . . . . . . 8 (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Abel)
543anim1i 1151 . . . . . . 7 ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
653expb 1119 . . . . . 6 ((𝐴 ∈ (SubRng‘𝑅) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑅 ∈ Abel ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)))
7 eqid 2735 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2735 . . . . . . 7 (+g𝑅) = (+g𝑅)
97, 8ablcom 19832 . . . . . 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 2824 . . . 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 3200 . 2 (𝐴 ∈ (SubRng‘𝑅) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(+g𝑅)𝑦) ∈ 𝐴 → (𝑦(+g𝑅)𝑥) ∈ 𝐴))
147, 8isnsg2 19187 . 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 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  SubGrpcsubg 19151  NrmSGrpcnsg 19152  Abelcabl 19814  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-nsg 19155  df-cmn 19815  df-abl 19816  df-rng 20171  df-subrng 20563
This theorem is referenced by:  rng2idlnsg  21294  rng2idlsubgnsg  21297
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