Theorem isnsg 18309

Description: Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
isnsg.1	⊢ 𝑋 = (Base‘𝐺)
isnsg.2	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
isnsg	⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isnsg

Dummy variables 𝑔 𝑏 𝑝 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nsg 18279	. . 3 ⊢ NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
2	1	mptrcl 6779	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3		subgrcl 18286	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
4	3	adantr 483	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
5		fveq2 6672	. . . . . 6 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6		fvexd 6687	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
7		fveq2 6672	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		isnsg.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	syl6eqr 2876	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
10		fvexd 6687	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) ∈ V)
11		simpl 485	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → 𝑔 = 𝐺)
12	11	fveq2d 6676	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = (+g‘𝐺))
13		isnsg.2	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
14	12, 13	syl6eqr 2876	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = + )
15		simplr 767	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
16		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
17	16	oveqd 7175	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
18	17	eleq1d 2899	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
19	16	oveqd 7175	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
20	19	eleq1d 2899	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑦𝑝𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠))
21	18, 20	bibi12d 348	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
22	15, 21	raleqbidv 3403	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
23	15, 22	raleqbidv 3403	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
24	10, 14, 23	sbcied2 3817	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → ([(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
25	6, 9, 24	sbcied2 3817	. . . . . 6 ⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
26	5, 25	rabeqbidv 3487	. . . . 5 ⊢ (𝑔 = 𝐺 → {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)} = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
27		fvex 6685	. . . . . 6 ⊢ (SubGrp‘𝐺) ∈ V
28	27	rabex 5237	. . . . 5 ⊢ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
29	26, 1, 28	fvmpt 6770	. . . 4 ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
30	29	eleq2d 2900	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
31		eleq2 2903	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
32		eleq2 2903	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑦 + 𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑆))
33	31, 32	bibi12d 348	. . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
34	33	2ralbidv 3201	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
35	34	elrab 3682	. . 3 ⊢ (𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
36	30, 35	syl6bb 289	. 2 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))))
37	2, 4, 36	pm5.21nii 382	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144  Vcvv 3496  [wsbc 3774  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  Grpcgrp 18105  SubGrpcsubg 18275  NrmSGrpcnsg 18276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-subg 18278  df-nsg 18279

This theorem is referenced by:  isnsg2  18310  nsgbi  18311  nsgsubg  18312  isnsg4  18321  nmznsg  18322  ablnsg  18969  rzgrp  20769