Theorem tngngp2 22456

Description: A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
tngngp2.t	⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
tngngp2.x	⊢ 𝑋 = (Base‘𝐺)
tngngp2.d	⊢ 𝐷 = (dist‘𝑇)

Assertion

Ref	Expression
tngngp2	⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Proof of Theorem tngngp2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ngpgrp 22403	. . . . 5 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2		tngngp2.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
3		fvex 6201	. . . . . . . 8 ⊢ (Base‘𝐺) ∈ V
4	2, 3	eqeltri 2697	. . . . . . 7 ⊢ 𝑋 ∈ V
5		reex 10027	. . . . . . 7 ⊢ ℝ ∈ V
6		fex2 7121	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
7	4, 5, 6	mp3an23 1416	. . . . . 6 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
8	2	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
9		tngngp2.t	. . . . . . . 8 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
10	9, 2	tngbas 22445	. . . . . . 7 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
11		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
12	9, 11	tngplusg 22446	. . . . . . . 8 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
13	12	oveqdr 6674	. . . . . . 7 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
14	8, 10, 13	grppropd 17437	. . . . . 6 ⊢ (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
15	7, 14	syl 17	. . . . 5 ⊢ (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
16	1, 15	syl5ibr 236	. . . 4 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
17	16	imp 445	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
18		ngpms 22404	. . . . . 6 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
19	18	adantl 482	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
20		eqid 2622	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
21		tngngp2.d	. . . . . 6 ⊢ 𝐷 = (dist‘𝑇)
22	20, 21	msmet2 22265	. . . . 5 ⊢ (𝑇 ∈ MetSp → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
23	19, 22	syl 17	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
24		eqid 2622	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
25	2, 24	grpsubf 17494	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋)
26	17, 25	syl 17	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋)
27		fco 6058	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
28	26, 27	syldan 487	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
29	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
30	9, 24	tngds 22452	. . . . . . . . . 10 ⊢ (𝑁 ∈ V → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
32	31, 21	syl6reqr 2675	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g‘𝐺)))
33	32	feq1d 6030	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ))
34	28, 33	mpbird 247	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
35		ffn 6045	. . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
36		fnresdm 6000	. . . . . 6 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
37	34, 35, 36	3syl 18	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
38	29, 10	syl 17	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑋 = (Base‘𝑇))
39	38	sqxpeqd 5141	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
40	39	reseq2d 5396	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
41	37, 40	eqtr3d 2658	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
42	38	fveq2d 6195	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
43	23, 41, 42	3eltr4d 2716	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
44	17, 43	jca 554	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
45	15	biimpa 501	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
46	45	adantrr 753	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
47		simprr 796	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
48	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
49	48, 10	syl 17	. . . . . . . . 9 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
50	49	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
51	47, 50	eleqtrd 2703	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
52		metf 22135	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
53	51, 52	syl 17	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
54		ffn 6045	. . . . . 6 ⊢ (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
55		fnresdm 6000	. . . . . 6 ⊢ (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
56	53, 54, 55	3syl 18	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
57	56, 51	eqeltrd 2701	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
58	56	fveq2d 6195	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
59		simprl 794	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
60		eqid 2622	. . . . . . 7 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
61	9, 21, 60	tngtopn 22454	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
62	59, 48, 61	syl2anc 693	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
63	58, 62	eqtr2d 2657	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
64		eqid 2622	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
65	21	reseq1i 5392	. . . . 5 ⊢ (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
66	64, 20, 65	isms2 22255	. . . 4 ⊢ (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
67	57, 63, 66	sylanbrc 698	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ MetSp)
68		simpl 473	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁:𝑋⟶ℝ)
69	9, 2, 5	tngnm 22455	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
70	59, 68, 69	syl2anc 693	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
71	8, 10	eqtr3d 2658	. . . . . . . 8 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
72	71, 12	grpsubpropd 17520	. . . . . . 7 ⊢ (𝑁 ∈ V → (-g‘𝐺) = (-g‘𝑇))
73	48, 72	syl 17	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g‘𝐺) = (-g‘𝑇))
74	70, 73	coeq12d 5286	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = ((norm‘𝑇) ∘ (-g‘𝑇)))
75	48, 30	syl 17	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
76	74, 75	eqtr3d 2658	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇))
77		eqimss 3657	. . . 4 ⊢ (((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
78	76, 77	syl 17	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
79		eqid 2622	. . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇)
80		eqid 2622	. . . 4 ⊢ (-g‘𝑇) = (-g‘𝑇)
81		eqid 2622	. . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇)
82	79, 80, 81	isngp 22400	. . 3 ⊢ (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇)))
83	46, 67, 78, 82	syl3anbrc 1246	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
84	44, 83	impbida 877	1 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574   × cxp 5112   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  Basecbs 15857  +_gcplusg 15941  distcds 15950  TopOpenctopn 16082  Grpcgrp 17422  -_gcsg 17424  Metcme 19732  MetOpencmopn 19736  MetSpcmt 22123  normcnm 22381  NrmGrpcngp 22382   toNrmGrp ctng 22383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-tset 15960  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-tng 22389

This theorem is referenced by:  tngngpd  22457  tngngp  22458  nrmtngnrm  22462  tngngpim  22463  tngnrg  22478