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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.
StepHypRef Expression
1 ngpgrp 23861 . . . . 5 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2 tngngp2.x . . . . . . . 8 𝑋 = (Base‘𝐺)
32fvexi 6839 . . . . . . 7 𝑋 ∈ V
4 reex 11063 . . . . . . 7 ℝ ∈ V
5 fex2 7848 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) → 𝑁 ∈ V)
63, 4, 5mp3an23 1452 . . . . . 6 (𝑁:𝑋⟶ℝ → 𝑁 ∈ V)
72a1i 11 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝐺))
8 tngngp2.t . . . . . . . 8 𝑇 = (𝐺 toNrmGrp 𝑁)
98, 2tngbas 23904 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10 eqid 2736 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
118, 10tngplusg 23906 . . . . . . . 8 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
1211oveqdr 7365 . . . . . . 7 ((𝑁 ∈ V ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
137, 9, 12grppropd 18690 . . . . . 6 (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
146, 13syl 17 . . . . 5 (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
151, 14syl5ibr 245 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
1615imp 407 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17 ngpms 23862 . . . . . 6 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
1817adantl 482 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19 eqid 2736 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
20 tngngp2.d . . . . . 6 𝐷 = (dist‘𝑇)
2119, 20msmet2 23719 . . . . 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 2736 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
242, 23grpsubf 18750 . . . . . . . . 9 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
2516, 24syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
26 fco 6675 . . . . . . . 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 23917 . . . . . . . . . 10 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3028, 29syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3120, 30eqtr4id 2795 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g𝐺)))
3231feq1d 6636 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ))
3327, 32mpbird 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)
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 5652 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
3938reseq2d 5923 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4036, 39eqtr3d 2778 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4137fveq2d 6829 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
4222, 40, 413eltr4d 2852 . . 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 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))
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 6829 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
5046, 49eleqtrd 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)
5250, 51syl 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)
5552, 53, 543syl 18 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5655, 50eqeltrd 2837 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
5755fveq2d 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)
608, 20, 59tngtopn 23920 . . . . . 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 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)
6420reseq1i 5919 . . . . 5 (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
6563, 19, 64isms2 23709 . . . 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 23921 . . . . . . 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 2778 . . . . . . . 8 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
7170, 11grpsubpropd 18776 . . . . . . 7 (𝑁 ∈ V → (-g𝐺) = (-g𝑇))
7247, 71syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g𝐺) = (-g𝑇))
7369, 72coeq12d 5806 . . . . 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 2778 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇))
76 eqimss 3988 . . . 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 2736 . . . 4 (norm‘𝑇) = (norm‘𝑇)
79 eqid 2736 . . . 4 (-g𝑇) = (-g𝑇)
80 eqid 2736 . . . 4 (dist‘𝑇) = (dist‘𝑇)
8178, 79, 80isngp 23858 . . 3 (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇)))
8245, 66, 77, 81syl3anbrc 1342 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
8343, 82impbida 798 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
Colors of variables: wff setvar class
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  cr 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
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