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Theorem ablnsg 19621
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 2736 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2736 . . . . . . 7 (+g𝐺) = (+g𝐺)
31, 2ablcom 19577 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
433expb 1120 . . . . 5 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g𝐺)𝑧) = (𝑧(+g𝐺)𝑦))
54eleq1d 2822 . . . 4 ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
65ralrimivva 3196 . . 3 (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥))
71, 2isnsg 18953 . . . 4 (𝑥 ∈ (NrmSGrp‘𝐺) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g𝐺)𝑦) ∈ 𝑥)))
87rbaib 539 . . 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 2734 1 (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3063  cfv 6494  (class class class)co 7354  Basecbs 17080  +gcplusg 17130  SubGrpcsubg 18918  NrmSGrpcnsg 18919  Abelcabl 19559
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fv 6502  df-ov 7357  df-subg 18921  df-nsg 18922  df-cmn 19560  df-abl 19561
This theorem is referenced by:  qusabl  19639  ablsimpnosubgd  19879  ablsimpgprmd  19890  qus1  20701  qusrhm  20703  quslmod  32041  quslmhm  32042  lidlnsg  32109  qusdimsum  32214
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