Theorem isnsg 13725

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 13694	. . 3 ⊢ NrmSGrp = (𝑔 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑔) ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)})
2	1	mptrcl 5710	. 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
3		subgrcl 13702	. . 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 5623	. . . . . 6 ⊢ (𝑔 = 𝐺 → (SubGrp‘𝑔) = (SubGrp‘𝐺))
6		basfn 13077	. . . . . . . . . 10 ⊢ Base Fn V
7		funfvex 5640	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑔 ∈ dom Base) → (Base‘𝑔) ∈ V)
8	7	funfni 5419	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑔 ∈ V) → (Base‘𝑔) ∈ V)
9	6, 8	mpan 424	. . . . . . . . 9 ⊢ (𝑔 ∈ V → (Base‘𝑔) ∈ V)
10	9	elv 2803	. . . . . . . 8 ⊢ (Base‘𝑔) ∈ V
11	10	a1i 9	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
12		fveq2 5623	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
13		isnsg.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
14	12, 13	eqtr4di 2280	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑋)
15		plusgslid 13131	. . . . . . . . . . 11 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ)
16	15	slotex 13045	. . . . . . . . . 10 ⊢ (𝑔 ∈ V → (+g‘𝑔) ∈ V)
17	16	elv 2803	. . . . . . . . 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 5627	. . . . . . . . 9 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = (+g‘𝐺))
21		isnsg.2	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
22	20, 21	eqtr4di 2280	. . . . . . . 8 ⊢ ((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) → (+g‘𝑔) = + )
23		simplr 528	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑏 = 𝑋)
24		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → 𝑝 = + )
25	24	oveqd 6011	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑥𝑝𝑦) = (𝑥 + 𝑦))
26	25	eleq1d 2298	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑠))
27	24	oveqd 6011	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (𝑦𝑝𝑥) = (𝑦 + 𝑥))
28	27	eleq1d 2298	. . . . . . . . . . 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 2744	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑏 = 𝑋) ∧ 𝑝 = + ) → (∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)))
31	23, 30	raleqbidv 2744	. . . . . . . 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 3066	. . . . . . 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 3066	. . . . . 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 2794	. . . . 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 13699	. . . . . 6 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)
37		rabexg 4226	. . . . . 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 5721	. . . 4 ⊢ (𝐺 ∈ Grp → (NrmSGrp‘𝐺) = {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)})
40	39	eleq2d 2299	. . 3 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (NrmSGrp‘𝐺) ↔ 𝑆 ∈ {𝑠 ∈ (SubGrp‘𝐺) ∣ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑦 + 𝑥) ∈ 𝑠)}))
41		eleq2 2293	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑥 + 𝑦) ∈ 𝑠 ↔ (𝑥 + 𝑦) ∈ 𝑆))
42		eleq2 2293	. . . . . 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 2554	. . . 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 2959	. . 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 709	1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2799  [wsbc 3028   Fn wfn 5309  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  +_gcplusg 13096  Grpcgrp 13519  SubGrpcsubg 13690  NrmSGrpcnsg 13691

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-ov 5997  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-subg 13693  df-nsg 13694

This theorem is referenced by:  isnsg2  13726  nsgbi  13727  nsgsubg  13728  isnsg4  13735  nmznsg  13736  ablnsg  13857