Theorem tngngp2 24694

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 24633	. . . . 5 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2		tngngp2.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
3	2	fvexi 6934	. . . . . . 7 ⊢ 𝑋 ∈ V
4		reex 11275	. . . . . . 7 ⊢ ℝ ∈ V
5		fex2 7974	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
6	3, 4, 5	mp3an23 1453	. . . . . 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 24676	. . . . . . 7 ⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10		eqid 2740	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
11	8, 10	tngplusg 24678	. . . . . . . 8 ⊢ (𝑁 ∈ V → (+g‘𝐺) = (+g‘𝑇))
12	11	oveqdr 7476	. . . . . . 7 ⊢ ((𝑁 ∈ V ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑇)𝑦))
13	7, 9, 12	grppropd 18991	. . . . . 6 ⊢ (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
14	6, 13	syl 17	. . . . 5 ⊢ (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
15	1, 14	imbitrrid 246	. . . 4 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
16	15	imp 406	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17		ngpms 24634	. . . . . 6 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
18	17	adantl 481	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19		eqid 2740	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
20		tngngp2.d	. . . . . 6 ⊢ 𝐷 = (dist‘𝑇)
21	19, 20	msmet2 24491	. . . . 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 2740	. . . . . . . . . 10 ⊢ (-g‘𝐺) = (-g‘𝐺)
24	2, 23	grpsubf 19059	. . . . . . . . 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 6771	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
27	25, 26	syldan 590	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ)
28	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
29	8, 23	tngds 24689	. . . . . . . . . 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 2799	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g‘𝐺)))
32	31	feq1d 6732	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g‘𝐺)):(𝑋 × 𝑋)⟶ℝ))
33	27, 32	mpbird 257	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
34		ffn 6747	. . . . . 6 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
35		fnresdm 6699	. . . . . 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 5732	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
39	38	reseq2d 6009	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
40	36, 39	eqtr3d 2782	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
41	37	fveq2d 6924	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
42	22, 40, 41	3eltr4d 2859	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
43	16, 42	jca 511	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
44	14	biimpa 476	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
45	44	adantrr 716	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
46		simprr 772	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
47	6	adantr 480	. . . . . . . . 9 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
48	47, 9	syl 17	. . . . . . . 8 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
49	48	fveq2d 6924	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
50	46, 49	eleqtrd 2846	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
51		metf 24361	. . . . . 6 ⊢ (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
52		ffn 6747	. . . . . 6 ⊢ (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
53		fnresdm 6699	. . . . . 6 ⊢ (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
54	50, 51, 52, 53	4syl 19	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
55	54, 50	eqeltrd 2844	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
56	54	fveq2d 6924	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
57		simprl 770	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
58		eqid 2740	. . . . . . 7 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
59	8, 20, 58	tngtopn 24692	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
60	57, 47, 59	syl2anc 583	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
61	56, 60	eqtr2d 2781	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
62		eqid 2740	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
63	20	reseq1i 6005	. . . . 5 ⊢ (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
64	62, 19, 63	isms2 24481	. . . 4 ⊢ (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
65	55, 61, 64	sylanbrc 582	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ MetSp)
66		simpl 482	. . . . . . 7 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁:𝑋⟶ℝ)
67	8, 2, 4	tngnm 24693	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
68	57, 66, 67	syl2anc 583	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
69	7, 9	eqtr3d 2782	. . . . . . . 8 ⊢ (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
70	69, 11	grpsubpropd 19085	. . . . . . 7 ⊢ (𝑁 ∈ V → (-g‘𝐺) = (-g‘𝑇))
71	47, 70	syl 17	. . . . . 6 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g‘𝐺) = (-g‘𝑇))
72	68, 71	coeq12d 5889	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = ((norm‘𝑇) ∘ (-g‘𝑇)))
73	47, 29	syl 17	. . . . 5 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇))
74	72, 73	eqtr3d 2782	. . . 4 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇))
75		eqimss 4067	. . . 4 ⊢ (((norm‘𝑇) ∘ (-g‘𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
76	74, 75	syl 17	. . 3 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇))
77		eqid 2740	. . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇)
78		eqid 2740	. . . 4 ⊢ (-g‘𝑇) = (-g‘𝑇)
79		eqid 2740	. . . 4 ⊢ (dist‘𝑇) = (dist‘𝑇)
80	77, 78, 79	isngp 24630	. . 3 ⊢ (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g‘𝑇)) ⊆ (dist‘𝑇)))
81	45, 65, 76, 80	syl3anbrc 1343	. 2 ⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
82	43, 81	impbida 800	1 ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   × cxp 5698   ↾ cres 5702   ∘ ccom 5704   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  Basecbs 17258  +_gcplusg 17311  distcds 17320  TopOpenctopn 17481  Grpcgrp 18973  -_gcsg 18975  Metcmet 21373  MetOpencmopn 21377  MetSpcms 24349  normcnm 24610  NrmGrpcngp 24611   toNrmGrp ctng 24612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-plusg 17324  df-tset 17330  df-ds 17333  df-rest 17482  df-topn 17483  df-0g 17501  df-topgen 17503  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-sbg 18978  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-xms 24351  df-ms 24352  df-nm 24616  df-ngp 24617  df-tng 24618

This theorem is referenced by:  tngngpd  24695  tngngp  24696  nrmtngnrm  24700  tngngpim  24701  tngnrg  24716