MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablnsg Structured version   Visualization version   GIF version

Theorem ablnsg 19865
Description: Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
Assertion
Ref Expression
ablnsg (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Proof of Theorem ablnsg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2737 . . . . . . 7 (+g𝐺) = (+g𝐺)
31, 2ablcom 19817 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
433expb 1121 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
54eleq1d 2826 . . . 4 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
65ralrimivva 3202 . . 3 (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
71, 2isnsg 19173 . . . 4 (𝑥 ∈ (NrmSGrp‘𝐺) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥)))
87rbaib 538 . . 3 (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥) → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺)))
96, 8syl 17 . 2 (𝐺 ∈ Abel → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺)))
109eqrdv 2735 1 (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  SubGrpcsubg 19138  NrmSGrpcnsg 19139  Abelcabl 19799
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-nsg 19142  df-cmn 19800  df-abl 19801
This theorem is referenced by:  qusabl  19883  ablsimpnosubgd  20124  ablsimpgprmd  20135  rngansg  20167  lidlnsg  21258  qus2idrng  21283  qus1  21284  qusrhm  21286  quslmod  33386  quslmhm  33387  qusdimsum  33679
  Copyright terms: Public domain W3C validator