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Theorem ablnsg 19035
 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 2758 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2758 . . . . . . 7 (+g𝐺) = (+g𝐺)
31, 2ablcom 18991 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
433expb 1117 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
54eleq1d 2836 . . . 4 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
65ralrimivva 3120 . . 3 (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
71, 2isnsg 18374 . . . 4 (𝑥 ∈ (NrmSGrp‘𝐺) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥)))
87rbaib 542 . . 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 2756 1 (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  SubGrpcsubg 18340  NrmSGrpcnsg 18341  Abelcabl 18974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-subg 18343  df-nsg 18344  df-cmn 18975  df-abl 18976 This theorem is referenced by:  qusabl  19053  ablsimpnosubgd  19294  ablsimpgprmd  19305  qus1  20076  qusrhm  20078  quslmod  31075  quslmhm  31076  lidlnsg  31142  qusdimsum  31230
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