Theorem tngngp2 23922

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 23861	. . . . 5 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2		tngngp2.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
3	2	fvexi 6839	. . . . . . 7 ⊢ 𝑋 ∈ V
4		reex 11063	. . . . . . 7 ⊢ ℝ ∈ V
5		fex2 7848	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
6	3, 4, 5	mp3an23 1452	. . . . . 6 ⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
7	2	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
8		tngngp2.t	. . . . . . . 8 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)
9	8, 2	tngbas 23904	. . . . . . 7 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
11	8, 10	tngplusg 23906	. . . . . . . 8 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
12	11	oveqdr 7365	. . . . . . 7 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
13	7, 9, 12	grppropd 18690	. . . . . 6 ⊢ (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
14	6, 13	syl 17	. . . . 5 ⊢ (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
15	1, 14	syl5ibr 245	. . . 4 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
16	15	imp 407	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17		ngpms 23862	. . . . . 6 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
18	17	adantl 482	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19		eqid 2736	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
20		tngngp2.d	. . . . . 6 ⊢ 𝐷 = (dist‘𝑇)
21	19, 20	msmet2 23719	. . . . 5 ⊢ (𝑇 ∈ MetSp → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
22	18, 21	syl 17	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
23		eqid 2736	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
24	2, 23	grpsubf 18750	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋)
25	16, 24	syl 17	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋)
26		fco 6675	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
27	25, 26	syldan 591	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
28	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
29	8, 23	tngds 23917	. . . . . . . . . 10 ⊢ (𝑁 ∈ V → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
30	28, 29	syl 17	. . . . . . . . 9 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
31	20, 30	eqtr4id 2795	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g‘𝐺)))
32	31	feq1d 6636	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ))
33	27, 32	mpbird 256	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
34		ffn 6651	. . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
35		fnresdm 6603	. . . . . 6 ⊢ (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
36	33, 34, 35	3syl 18	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
37	28, 9	syl 17	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑋 = (Base‘𝑇))
38	37	sqxpeqd 5652	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
39	38	reseq2d 5923	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
40	36, 39	eqtr3d 2778	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
41	37	fveq2d 6829	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
42	22, 40, 41	3eltr4d 2852	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
43	16, 42	jca 512	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
44	14	biimpa 477	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
45	44	adantrr 714	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
46		simprr 770	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
47	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
48	47, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
49	48	fveq2d 6829	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
50	46, 49	eleqtrd 2839	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
51		metf 23589	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
52	50, 51	syl 17	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
53		ffn 6651	. . . . . 6 ⊢ (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
54		fnresdm 6603	. . . . . 6 ⊢ (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
55	52, 53, 54	3syl 18	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
56	55, 50	eqeltrd 2837	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
57	55	fveq2d 6829	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
58		simprl 768	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
59		eqid 2736	. . . . . . 7 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
60	8, 20, 59	tngtopn 23920	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
61	58, 47, 60	syl2anc 584	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
62	57, 61	eqtr2d 2777	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
63		eqid 2736	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
64	20	reseq1i 5919	. . . . 5 ⊢ (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
65	63, 19, 64	isms2 23709	. . . 4 ⊢ (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
66	56, 62, 65	sylanbrc 583	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ MetSp)
67		simpl 483	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁:𝑋⟶ℝ)
68	8, 2, 4	tngnm 23921	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
69	58, 67, 68	syl2anc 584	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
70	7, 9	eqtr3d 2778	. . . . . . . 8 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
71	70, 11	grpsubpropd 18776	. . . . . . 7 ⊢ (𝑁 ∈ V → (-g‘𝐺) = (-g‘𝑇))
72	47, 71	syl 17	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g‘𝐺) = (-g‘𝑇))
73	69, 72	coeq12d 5806	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = ((norm‘𝑇) ∘ (-g‘𝑇)))
74	47, 29	syl 17	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
75	73, 74	eqtr3d 2778	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇))
76		eqimss 3988	. . . 4 ⊢ (((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
77	75, 76	syl 17	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
78		eqid 2736	. . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇)
79		eqid 2736	. . . 4 ⊢ (-g‘𝑇) = (-g‘𝑇)
80		eqid 2736	. . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇)
81	78, 79, 80	isngp 23858	. . 3 ⊢ (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇)))
82	45, 66, 77, 81	syl3anbrc 1342	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
83	43, 82	impbida 798	1 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441   ⊆ wss 3898   × cxp 5618   ↾ cres 5622   ∘ ccom 5624   Fn wfn 6474  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337  ℝcr 10971  Basecbs 17009  +_gcplusg 17059  distcds 17068  TopOpenctopn 17229  Grpcgrp 18673  -_gcsg 18675  Metcmet 20689  MetOpencmopn 20693  MetSpcms 23577  normcnm 23838  NrmGrpcngp 23839   toNrmGrp ctng 23840

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-z 12421  df-dec 12539  df-uz 12684  df-q 12790  df-rp 12832  df-xneg 12949  df-xadd 12950  df-xmul 12951  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-plusg 17072  df-tset 17078  df-ds 17081  df-rest 17230  df-topn 17231  df-0g 17249  df-topgen 17251  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-minusg 18677  df-sbg 18678  df-psmet 20695  df-xmet 20696  df-met 20697  df-bl 20698  df-mopn 20699  df-top 22149  df-topon 22166  df-topsp 22188  df-bases 22202  df-xms 23579  df-ms 23580  df-nm 23844  df-ngp 23845  df-tng 23846

This theorem is referenced by:  tngngpd  23923  tngngp  23924  nrmtngnrm  23928  tngngpim  23929  tngnrg  23944