Theorem isnsg 19209

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 19178	. . 3 ⊢ NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
2	1	mptrcl 6989	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3		subgrcl 19185	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
4	3	adantr 485	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
5		fveq2 6871	. . . . . 6 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6		fvexd 6886	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
7		fveq2 6871	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
8		isnsg.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2818	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
10		fvexd 6886	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) ∈ V)
11		simpl 487	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → 𝑔 = 𝐺)
12	11	fveq2d 6875	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = (+g‘𝐺))
13		isnsg.2	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
14	12, 13	eqtr4di 2818	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = + )
15		simplr 780	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
16		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
17	16	oveqd 7417	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
18	17	eleq1d 2850	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
19	16	oveqd 7417	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
20	19	eleq1d 2850	. . . . . . . . . . 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 3339	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
23	15, 22	raleqbidv 3339	. . . . . . . 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 3791	. . . . . . 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 3791	. . . . . 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 3435	. . . . 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 6884	. . . . . 6 ⊢ (SubGrp‘𝐺) ∈ V
28	27	rabex 5299	. . . . 5 ⊢ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V
29	26, 1, 28	fvmpt 6979	. . . 4 ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
30	29	eleq2d 2851	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
31		eleq2 2854	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
32		eleq2 2854	. . . . . 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 3229	. . . 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 3653	. . 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	bitrdi 290	. 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 381	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417  Vcvv 3457  [wsbc 3747  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  +_gcplusg 17298  Grpcgrp 18988  SubGrpcsubg 19174  NrmSGrpcnsg 19175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  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 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-subg 19177  df-nsg 19178

This theorem is referenced by:  isnsg2  19210  nsgbi  19211  nsgsubg  19212  isnsg4  19221  nmznsg  19222  ablnsg  19905  rzgrp  21730