Theorem ablnsg 20112

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 2763	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ablcom 20064	. . . . . 6 ⊢ ((𝐺 ∈ Abel ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
4	3	3expb 1138	. . . . 5 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑦(+g‘𝐺)𝑧) = (𝑧(+g‘𝐺)𝑦))
5	4	eleq1d 2849	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
6	5	ralrimivva 3209	. . 3 ⊢ (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
7	1, 2	isnsg 19415	. . . 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 548	. . 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 18	. 2 ⊢ (𝐺 ∈ Abel → (𝑥 ∈ (NrmSGrp‘𝐺) ↔ 𝑥 ∈ (SubGrp‘𝐺)))
10	9	eqrdv 2761	1 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  SubGrpcsubg 19380  NrmSGrpcnsg 19381  Abelcabl 20046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-subg 19383  df-nsg 19384  df-cmn 20047  df-abl 20048

This theorem is used by:  qusabl  20130  ablsimpnosubgd  20371  ablsimpgprmd  20382  rngansg  20443  lidlnsg  21587  qus2idrng  21618  qus1  21619  qusrhm  21621  quslmod  34023  quslmhm  34024  qusdimsum  34371