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Theorem tngngp2 22944
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 22891 . . . . 5 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2 tngngp2.x . . . . . . . 8 𝑋 = (Base‘𝐺)
32fvexi 6555 . . . . . . 7 𝑋 ∈ V
4 reex 10477 . . . . . . 7 ℝ ∈ V
5 fex2 7497 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
63, 4, 5mp3an23 1445 . . . . . 6 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
72a1i 11 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
8 tngngp2.t . . . . . . . 8 𝑇 = (𝐺 toNrmGrp 𝑁)
98, 2tngbas 22933 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10 eqid 2794 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
118, 10tngplusg 22934 . . . . . . . 8 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
1211oveqdr 7047 . . . . . . 7 ((𝑁 ∈ V ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
137, 9, 12grppropd 17876 . . . . . 6 (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
146, 13syl 17 . . . . 5 (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
151, 14syl5ibr 247 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
1615imp 407 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17 ngpms 22892 . . . . . 6 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
1817adantl 482 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19 eqid 2794 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
20 tngngp2.d . . . . . 6 𝐷 = (dist‘𝑇)
2119, 20msmet2 22753 . . . . 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 2794 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
242, 23grpsubf 17935 . . . . . . . . 9 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
2516, 24syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
26 fco 6402 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (-g𝐺):(𝑋 × 𝑋)⟶𝑋) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
2725, 26syldan 591 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ)
286adantr 481 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
298, 23tngds 22940 . . . . . . . . . 10 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3028, 29syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3130, 20syl6reqr 2849 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g𝐺)))
3231feq1d 6370 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ))
3327, 32mpbird 258 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
34 ffn 6385 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
35 fnresdm 6339 . . . . . 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 5478 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
3938reseq2d 5737 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4036, 39eqtr3d 2832 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4137fveq2d 6545 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
4222, 40, 413eltr4d 2897 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
4316, 42jca 512 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
4414biimpa 477 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
4544adantrr 713 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
46 simprr 769 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
476adantr 481 . . . . . . . . . 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 6545 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
5046, 49eleqtrd 2884 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
51 metf 22623 . . . . . . 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 6385 . . . . . 6 (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
54 fnresdm 6339 . . . . . 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 2882 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
5755fveq2d 6545 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
58 simprl 767 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
59 eqid 2794 . . . . . . 7 (MetOpen‘𝐷) = (MetOpen‘𝐷)
608, 20, 59tngtopn 22942 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6158, 47, 60syl2anc 584 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6257, 61eqtr2d 2831 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
63 eqid 2794 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
6420reseq1i 5733 . . . . 5 (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
6563, 19, 64isms2 22743 . . . 4 (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
6656, 62, 65sylanbrc 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)
688, 2, 4tngnm 22943 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
6958, 67, 68syl2anc 584 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
707, 9eqtr3d 2832 . . . . . . . 8 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
7170, 11grpsubpropd 17961 . . . . . . 7 (𝑁 ∈ V → (-g𝐺) = (-g𝑇))
7247, 71syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g𝐺) = (-g𝑇))
7369, 72coeq12d 5624 . . . . 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 2832 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇))
76 eqimss 3946 . . . 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 2794 . . . 4 (norm‘𝑇) = (norm‘𝑇)
79 eqid 2794 . . . 4 (-g𝑇) = (-g𝑇)
80 eqid 2794 . . . 4 (dist‘𝑇) = (dist‘𝑇)
8178, 79, 80isngp 22888 . . 3 (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇)))
8245, 66, 77, 81syl3anbrc 1336 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
8343, 82impbida 797 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2080  Vcvv 3436  wss 3861   × cxp 5444  cres 5448  ccom 5450   Fn wfn 6223  wf 6224  cfv 6228  (class class class)co 7019  cr 10385  Basecbs 16312  +gcplusg 16394  distcds 16403  TopOpenctopn 16524  Grpcgrp 17861  -gcsg 17863  Metcmet 20213  MetOpencmopn 20217  MetSpcms 22611  normcnm 22869  NrmGrpcngp 22870   toNrmGrp ctng 22871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-rep 5084  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322  ax-cnex 10442  ax-resscn 10443  ax-1cn 10444  ax-icn 10445  ax-addcl 10446  ax-addrcl 10447  ax-mulcl 10448  ax-mulrcl 10449  ax-mulcom 10450  ax-addass 10451  ax-mulass 10452  ax-distr 10453  ax-i2m1 10454  ax-1ne0 10455  ax-1rid 10456  ax-rnegex 10457  ax-rrecex 10458  ax-cnre 10459  ax-pre-lttri 10460  ax-pre-lttrn 10461  ax-pre-ltadd 10462  ax-pre-mulgt0 10463  ax-pre-sup 10464
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-nel 3090  df-ral 3109  df-rex 3110  df-reu 3111  df-rmo 3112  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-pss 3878  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-tp 4479  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-tr 5067  df-id 5351  df-eprel 5356  df-po 5365  df-so 5366  df-fr 5405  df-we 5407  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-pred 6026  df-ord 6072  df-on 6073  df-lim 6074  df-suc 6075  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-f1 6233  df-fo 6234  df-f1o 6235  df-fv 6236  df-riota 6980  df-ov 7022  df-oprab 7023  df-mpo 7024  df-om 7440  df-1st 7548  df-2nd 7549  df-wrecs 7801  df-recs 7863  df-rdg 7901  df-er 8142  df-map 8261  df-en 8361  df-dom 8362  df-sdom 8363  df-sup 8755  df-inf 8756  df-pnf 10526  df-mnf 10527  df-xr 10528  df-ltxr 10529  df-le 10530  df-sub 10721  df-neg 10722  df-div 11148  df-nn 11489  df-2 11550  df-3 11551  df-4 11552  df-5 11553  df-6 11554  df-7 11555  df-8 11556  df-9 11557  df-n0 11748  df-z 11832  df-dec 11949  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-plusg 16407  df-tset 16413  df-ds 16416  df-rest 16525  df-topn 16526  df-0g 16544  df-topgen 16546  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-xms 22613  df-ms 22614  df-nm 22875  df-ngp 22876  df-tng 22877
This theorem is referenced by:  tngngpd  22945  tngngp  22946  nrmtngnrm  22950  tngngpim  22951  tngnrg  22966
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