Theorem ablnsg 19845

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.

Step	Hyp	Ref	Expression
1		eqid 2726	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2726	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ablcom 19797	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
4	3	3expb 1117	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
5	4	eleq1d 2811	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
6	5	ralrimivva 3191	. . 3 ⊢ (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
7	1, 2	isnsg 19149	. . . 4 ⊢ (𝑥 ∈ (NrmSGrp‘𝐺) ↔ (𝑥 ∈ (SubGrp‘𝐺) ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥)))
8	7	rbaib 537	. . 3 ⊢ (∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥) → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺)))
9	6, 8	syl 17	. 2 ⊢ (𝐺 ∈ Abel → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺)))
10	9	eqrdv 2724	1 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  +_gcplusg 17266  SubGrpcsubg 19114  NrmSGrpcnsg 19115  Abelcabl 19779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-subg 19117  df-nsg 19118  df-cmn 19780  df-abl 19781

This theorem is referenced by:  qusabl  19863  ablsimpnosubgd  20104  ablsimpgprmd  20115  rngansg  20153  lidlnsg  21237  qus2idrng  21262  qus1  21263  qusrhm  21265  quslmod  33233  quslmhm  33234  qusdimsum  33523