Theorem ablnsg 20070

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 2760	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2760	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
3	1, 2	ablcom 20022	. . . . . 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 2845	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ (𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
6	5	ralrimivva 3205	. . 3 ⊢ (𝐺 ∈ Abel → ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ∈ 𝑥 ↔ (𝑧(+g‘𝐺)𝑦) ∈ 𝑥))
7	1, 2	isnsg 19374	. . . 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 2758	1 ⊢ (𝐺 ∈ Abel → (NrmSGrp‘𝐺) = (SubGrp‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  SubGrpcsubg 19339  NrmSGrpcnsg 19340  Abelcabl 20004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-subg 19342  df-nsg 19343  df-cmn 20005  df-abl 20006

This theorem is used by:  qusabl  20088  ablsimpnosubgd  20329  ablsimpgprmd  20340  rngansg  20401  lidlnsg  21545  qus2idrng  21576  qus1  21577  qusrhm  21579  quslmod  33969  quslmhm  33970  qusdimsum  34310