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Theorem tngngp2 22781
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 22728 . . . . 5 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2 tngngp2.x . . . . . . . 8 𝑋 = (Base‘𝐺)
32fvexi 6423 . . . . . . 7 𝑋 ∈ V
4 reex 10313 . . . . . . 7 ℝ ∈ V
5 fex2 7354 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
63, 4, 5mp3an23 1578 . . . . . 6 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
72a1i 11 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
8 tngngp2.t . . . . . . . 8 𝑇 = (𝐺 toNrmGrp 𝑁)
98, 2tngbas 22770 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10 eqid 2797 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
118, 10tngplusg 22771 . . . . . . . 8 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
1211oveqdr 6904 . . . . . . 7 ((𝑁 ∈ V ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
137, 9, 12grppropd 17750 . . . . . 6 (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
146, 13syl 17 . . . . 5 (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
151, 14syl5ibr 238 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
1615imp 396 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17 ngpms 22729 . . . . . 6 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
1817adantl 474 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19 eqid 2797 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
20 tngngp2.d . . . . . 6 𝐷 = (dist‘𝑇)
2119, 20msmet2 22590 . . . . 5 (𝑇 ∈ MetSp → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
2218, 21syl 17 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
23 eqid 2797 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
242, 23grpsubf 17807 . . . . . . . . 9 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
2516, 24syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
26 fco 6271 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (-g𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
2725, 26syldan 586 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
286adantr 473 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
298, 23tngds 22777 . . . . . . . . . 10 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3028, 29syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3130, 20syl6reqr 2850 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g𝐺)))
3231feq1d 6239 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ))
3327, 32mpbird 249 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
34 ffn 6254 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
35 fnresdm 6209 . . . . . 6 (𝐷 Fn (𝑋 × 𝑋) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
3633, 34, 353syl 18 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = 𝐷)
3728, 9syl 17 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑋 = (Base‘𝑇))
3837sqxpeqd 5342 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
3938reseq2d 5598 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4036, 39eqtr3d 2833 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4137fveq2d 6413 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
4222, 40, 413eltr4d 2891 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
4316, 42jca 508 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
4414biimpa 469 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
4544adantrr 709 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
46 simprr 790 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
476adantr 473 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
4847, 9syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
4948fveq2d 6413 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
5046, 49eleqtrd 2878 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
51 metf 22460 . . . . . . 7 (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
5250, 51syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
53 ffn 6254 . . . . . 6 (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
54 fnresdm 6209 . . . . . 6 (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5552, 53, 543syl 18 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5655, 50eqeltrd 2876 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
5755fveq2d 6413 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
58 simprl 788 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
59 eqid 2797 . . . . . . 7 (MetOpen‘𝐷) = (MetOpen‘𝐷)
608, 20, 59tngtopn 22779 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6158, 47, 60syl2anc 580 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6257, 61eqtr2d 2832 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
63 eqid 2797 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
6420reseq1i 5594 . . . . 5 (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
6563, 19, 64isms2 22580 . . . 4 (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
6656, 62, 65sylanbrc 579 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ MetSp)
67 simpl 475 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁:𝑋⟶ℝ)
688, 2, 4tngnm 22780 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
6958, 67, 68syl2anc 580 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
707, 9eqtr3d 2833 . . . . . . . 8 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
7170, 11grpsubpropd 17833 . . . . . . 7 (𝑁 ∈ V → (-g𝐺) = (-g𝑇))
7247, 71syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g𝐺) = (-g𝑇))
7369, 72coeq12d 5488 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = ((norm‘𝑇) ∘ (-g𝑇)))
7447, 29syl 17 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
7573, 74eqtr3d 2833 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇))
76 eqimss 3851 . . . 4 (((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
7775, 76syl 17 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
78 eqid 2797 . . . 4 (norm‘𝑇) = (norm‘𝑇)
79 eqid 2797 . . . 4 (-g𝑇) = (-g𝑇)
80 eqid 2797 . . . 4 (dist‘𝑇) = (dist‘𝑇)
8178, 79, 80isngp 22725 . . 3 (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇)))
8245, 66, 77, 81syl3anbrc 1444 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
8343, 82impbida 836 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3383  wss 3767   × cxp 5308  cres 5312  ccom 5314   Fn wfn 6094  wf 6095  cfv 6099  (class class class)co 6876  cr 10221  Basecbs 16181  +gcplusg 16264  distcds 16273  TopOpenctopn 16394  Grpcgrp 17735  -gcsg 17737  Metcmet 20051  MetOpencmopn 20055  MetSpcms 22448  normcnm 22706  NrmGrpcngp 22707   toNrmGrp ctng 22708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299  ax-pre-sup 10300
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-er 7980  df-map 8095  df-en 8194  df-dom 8195  df-sdom 8196  df-sup 8588  df-inf 8589  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-div 10975  df-nn 11311  df-2 11372  df-3 11373  df-4 11374  df-5 11375  df-6 11376  df-7 11377  df-8 11378  df-9 11379  df-n0 11577  df-z 11663  df-dec 11780  df-uz 11927  df-q 12030  df-rp 12071  df-xneg 12189  df-xadd 12190  df-xmul 12191  df-ndx 16184  df-slot 16185  df-base 16187  df-sets 16188  df-plusg 16277  df-tset 16283  df-ds 16286  df-rest 16395  df-topn 16396  df-0g 16414  df-topgen 16416  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-sbg 17740  df-psmet 20057  df-xmet 20058  df-met 20059  df-bl 20060  df-mopn 20061  df-top 21024  df-topon 21041  df-topsp 21063  df-bases 21076  df-xms 22450  df-ms 22451  df-nm 22712  df-ngp 22713  df-tng 22714
This theorem is referenced by:  tngngpd  22782  tngngp  22783  nrmtngnrm  22787  tngngpim  22788  tngnrg  22803
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