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Theorem tngngp2 22366
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.
StepHypRef Expression
1 ngpgrp 22313 . . . . 5 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2 tngngp2.x . . . . . . . 8 𝑋 = (Base‘𝐺)
3 fvex 6158 . . . . . . . 8 (Base‘𝐺) ∈ V
42, 3eqeltri 2694 . . . . . . 7 𝑋 ∈ V
5 reex 9971 . . . . . . 7 ℝ ∈ V
6 fex2 7068 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
74, 5, 6mp3an23 1413 . . . . . 6 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
82a1i 11 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
9 tngngp2.t . . . . . . . 8 𝑇 = (𝐺 toNrmGrp 𝑁)
109, 2tngbas 22355 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
11 eqid 2621 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
129, 11tngplusg 22356 . . . . . . . 8 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
1312oveqdr 6628 . . . . . . 7 ((𝑁 ∈ V ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
148, 10, 13grppropd 17358 . . . . . 6 (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
157, 14syl 17 . . . . 5 (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
161, 15syl5ibr 236 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
1716imp 445 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
18 ngpms 22314 . . . . . 6 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
1918adantl 482 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
20 eqid 2621 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
21 tngngp2.d . . . . . 6 𝐷 = (dist‘𝑇)
2220, 21msmet2 22175 . . . . 5 (𝑇 ∈ MetSp → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
2319, 22syl 17 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
24 eqid 2621 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
252, 24grpsubf 17415 . . . . . . . . 9 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
2617, 25syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
27 fco 6015 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (-g𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
2826, 27syldan 487 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
297adantr 481 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
309, 24tngds 22362 . . . . . . . . . 10 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3129, 30syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3231, 21syl6reqr 2674 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g𝐺)))
3332feq1d 5987 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ))
3428, 33mpbird 247 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
35 ffn 6002 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
36 fnresdm 5958 . . . . . 6 (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
3734, 35, 363syl 18 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
3829, 10syl 17 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑋 = (Base‘𝑇))
3938sqxpeqd 5101 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
4039reseq2d 5356 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4137, 40eqtr3d 2657 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4238fveq2d 6152 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
4323, 41, 423eltr4d 2713 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
4417, 43jca 554 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
4515biimpa 501 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
4645adantrr 752 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
47 simprr 795 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
487adantr 481 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
4948, 10syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
5049fveq2d 6152 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
5147, 50eleqtrd 2700 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
52 metf 22045 . . . . . . 7 (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
5351, 52syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
54 ffn 6002 . . . . . 6 (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
55 fnresdm 5958 . . . . . 6 (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5653, 54, 553syl 18 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5756, 51eqeltrd 2698 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
5856fveq2d 6152 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
59 simprl 793 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
60 eqid 2621 . . . . . . 7 (MetOpen‘𝐷) = (MetOpen‘𝐷)
619, 21, 60tngtopn 22364 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6259, 48, 61syl2anc 692 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6358, 62eqtr2d 2656 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
64 eqid 2621 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
6521reseq1i 5352 . . . . 5 (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
6664, 20, 65isms2 22165 . . . 4 (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
6757, 63, 66sylanbrc 697 . . 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)
699, 2, 5tngnm 22365 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
7059, 68, 69syl2anc 692 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
718, 10eqtr3d 2657 . . . . . . . 8 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
7271, 12grpsubpropd 17441 . . . . . . 7 (𝑁 ∈ V → (-g𝐺) = (-g𝑇))
7348, 72syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g𝐺) = (-g𝑇))
7470, 73coeq12d 5246 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = ((norm‘𝑇) ∘ (-g𝑇)))
7548, 30syl 17 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
7674, 75eqtr3d 2657 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇))
77 eqimss 3636 . . . 4 (((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
7876, 77syl 17 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
79 eqid 2621 . . . 4 (norm‘𝑇) = (norm‘𝑇)
80 eqid 2621 . . . 4 (-g𝑇) = (-g𝑇)
81 eqid 2621 . . . 4 (dist‘𝑇) = (dist‘𝑇)
8279, 80, 81isngp 22310 . . 3 (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇)))
8346, 67, 78, 82syl3anbrc 1244 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
8444, 83impbida 876 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555   × cxp 5072  cres 5076  ccom 5078   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cr 9879  Basecbs 15781  +gcplusg 15862  distcds 15871  TopOpenctopn 16003  Grpcgrp 17343  -gcsg 17345  Metcme 19651  MetOpencmopn 19655  MetSpcmt 22033  normcnm 22291  NrmGrpcngp 22292   toNrmGrp ctng 22293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-tset 15881  df-ds 15885  df-rest 16004  df-topn 16005  df-0g 16023  df-topgen 16025  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-xms 22035  df-ms 22036  df-nm 22297  df-ngp 22298  df-tng 22299
This theorem is referenced by:  tngngpd  22367  tngngp  22368  nrmtngnrm  22372  tngngpim  22373  tngnrg  22388
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