Theorem isnsg 13794

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 13763	. . 3 ⊢ NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
2	1	mptrcl 5729	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3		subgrcl 13771	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
4	3	adantr 276	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)) → 𝐺 ∈ Grp)
5		fveq2 5639	. . . . . 6 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6		basfn 13146	. . . . . . . . . 10 ⊢ Base Fn V
7		funfvex 5656	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
8	7	funfni 5432	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
9	6, 8	mpan 424	. . . . . . . . 9 ⊢ (𝑔 ∈ V → (Base‘𝑔) ∈ V)
10	9	elv 2806	. . . . . . . 8 ⊢ (Base‘𝑔) ∈ V
11	10	a1i 9	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
12		fveq2 5639	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
13		isnsg.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
14	12, 13	eqtr4di 2282	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
15		plusgslid 13200	. . . . . . . . . . 11 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
16	15	slotex 13114	. . . . . . . . . 10 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
17	16	elv 2806	. . . . . . . . 9 ⊢ (+g‘𝑔) ∈ V
18	17	a1i 9	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) ∈ V)
19		simpl 109	. . . . . . . . . 10 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → 𝑔 = 𝐺)
20	19	fveq2d 5643	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = (+g‘𝐺))
21		isnsg.2	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
22	20, 21	eqtr4di 2282	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = + )
23		simplr 529	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
24		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
25	24	oveqd 6035	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
26	25	eleq1d 2300	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
27	24	oveqd 6035	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
28	27	eleq1d 2300	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑦𝑝𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠))
29	26, 28	bibi12d 235	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
30	23, 29	raleqbidv 2746	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
31	23, 30	raleqbidv 2746	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
32	18, 22, 31	sbcied2 3069	. . . . . . 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)))
33	11, 14, 32	sbcied2 3069	. . . . . 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)))
34	5, 33	rabeqbidv 2797	. . . . 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)})
35		id 19	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp)
36		subgex 13768	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)
37		rabexg 4233	. . . . . 6 ⊢ ((SubGrp‘𝐺) ∈ V → {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V)
38	36, 37	syl 14	. . . . 5 ⊢ (𝐺 ∈ Grp → {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)} ∈ V)
39	1, 34, 35, 38	fvmptd3 5740	. . . 4 ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
40	39	eleq2d 2301	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
41		eleq2 2295	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
42		eleq2 2295	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑦 + 𝑥) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑆))
43	41, 42	bibi12d 235	. . . . 5 ⊢ (𝑠 = 𝑆 → (((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
44	43	2ralbidv 2556	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))
45	44	elrab 2962	. . 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)))
46	40, 45	bitrdi 196	. 2 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆))))
47	2, 4, 46	pm5.21nii 711	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802  [wsbc 3031   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018  Basecbs 13087  +_gcplusg 13165  Grpcgrp 13588  SubGrpcsubg 13759  NrmSGrpcnsg 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-inn 9144  df-2 9202  df-ndx 13090  df-slot 13091  df-base 13093  df-plusg 13178  df-subg 13762  df-nsg 13763

This theorem is referenced by:  isnsg2  13795  nsgbi  13796  nsgsubg  13797  isnsg4  13804  nmznsg  13805  ablnsg  13926