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Theorem tngngp2 24688
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 24627 . . . . 5 (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp)
2 tngngp2.x . . . . . . . 8 𝑋 = (Base‘𝐺)
32fvexi 6920 . . . . . . 7 𝑋 ∈ V
4 reex 11243 . . . . . . 7 ℝ ∈ V
5 fex2 7956 . . . . . . 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 24670 . . . . . . 7 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
10 eqid 2734 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
118, 10tngplusg 24672 . . . . . . . 8 (𝑁 ∈ V → (+g𝐺) = (+g𝑇))
1211oveqdr 7458 . . . . . . 7 ((𝑁 ∈ V ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝑇)𝑦))
137, 9, 12grppropd 18981 . . . . . 6 (𝑁 ∈ V → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
146, 13syl 17 . . . . 5 (𝑁:𝑋⟶ℝ → (𝐺 ∈ Grp ↔ 𝑇 ∈ Grp))
151, 14imbitrrid 246 . . . 4 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp → 𝐺 ∈ Grp))
1615imp 406 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
17 ngpms 24628 . . . . . 6 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
1817adantl 481 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ MetSp)
19 eqid 2734 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
20 tngngp2.d . . . . . 6 𝐷 = (dist‘𝑇)
2119, 20msmet2 24485 . . . . 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 2734 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
242, 23grpsubf 19049 . . . . . . . . 9 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
2516, 24syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
26 fco 6760 . . . . . . . 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 480 . . . . . . . . . 10 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝑁 ∈ V)
298, 23tngds 24683 . . . . . . . . . 10 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3028, 29syl 17 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3120, 30eqtr4id 2793 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝑁 ∘ (-g𝐺)))
3231feq1d 6720 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ (𝑁 ∘ (-g𝐺)):(𝑋 × 𝑋)⟶ℝ))
3327, 32mpbird 257 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷:(𝑋 × 𝑋)⟶ℝ)
34 ffn 6736 . . . . . 6 (𝐷:(𝑋 × 𝑋)⟶ℝ → 𝐷 Fn (𝑋 × 𝑋))
35 fnresdm 6687 . . . . . 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 5720 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝑋 × 𝑋) = ((Base‘𝑇) × (Base‘𝑇)))
3938reseq2d 5999 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4036, 39eqtr3d 2776 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 = (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))
4137fveq2d 6910 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
4222, 40, 413eltr4d 2853 . . 3 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐷 ∈ (Met‘𝑋))
4316, 42jca 511 . 2 ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋)))
4414biimpa 476 . . . 4 ((𝑁:𝑋⟶ℝ ∧ 𝐺 ∈ Grp) → 𝑇 ∈ Grp)
4544adantrr 717 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ Grp)
46 simprr 773 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘𝑋))
476adantr 480 . . . . . . . . 9 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 ∈ V)
4847, 9syl 17 . . . . . . . 8 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑋 = (Base‘𝑇))
4948fveq2d 6910 . . . . . . 7 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (Met‘𝑋) = (Met‘(Base‘𝑇)))
5046, 49eleqtrd 2840 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐷 ∈ (Met‘(Base‘𝑇)))
51 metf 24355 . . . . . 6 (𝐷 ∈ (Met‘(Base‘𝑇)) → 𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ)
52 ffn 6736 . . . . . 6 (𝐷:((Base‘𝑇) × (Base‘𝑇))⟶ℝ → 𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)))
53 fnresdm 6687 . . . . . 6 (𝐷 Fn ((Base‘𝑇) × (Base‘𝑇)) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5450, 51, 52, 534syl 19 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = 𝐷)
5554, 50eqeltrd 2838 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)))
5654fveq2d 6910 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘𝐷))
57 simprl 771 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝐺 ∈ Grp)
58 eqid 2734 . . . . . . 7 (MetOpen‘𝐷) = (MetOpen‘𝐷)
598, 20, 58tngtopn 24686 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑁 ∈ V) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6057, 47, 59syl2anc 584 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (MetOpen‘𝐷) = (TopOpen‘𝑇))
6156, 60eqtr2d 2775 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇)))))
62 eqid 2734 . . . . 5 (TopOpen‘𝑇) = (TopOpen‘𝑇)
6320reseq1i 5995 . . . . 5 (𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
6462, 19, 63isms2 24475 . . . 4 (𝑇 ∈ MetSp ↔ ((𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (Met‘(Base‘𝑇)) ∧ (TopOpen‘𝑇) = (MetOpen‘(𝐷 ↾ ((Base‘𝑇) × (Base‘𝑇))))))
6555, 61, 64sylanbrc 583 . . 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)
678, 2, 4tngnm 24687 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇))
6857, 66, 67syl2anc 584 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑁 = (norm‘𝑇))
697, 9eqtr3d 2776 . . . . . . . 8 (𝑁 ∈ V → (Base‘𝐺) = (Base‘𝑇))
7069, 11grpsubpropd 19075 . . . . . . 7 (𝑁 ∈ V → (-g𝐺) = (-g𝑇))
7147, 70syl 17 . . . . . 6 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (-g𝐺) = (-g𝑇))
7268, 71coeq12d 5877 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = ((norm‘𝑇) ∘ (-g𝑇)))
7347, 29syl 17 . . . . 5 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
7472, 73eqtr3d 2776 . . . 4 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇))
75 eqimss 4053 . . . 4 (((norm‘𝑇) ∘ (-g𝑇)) = (dist‘𝑇) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
7674, 75syl 17 . . 3 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇))
77 eqid 2734 . . . 4 (norm‘𝑇) = (norm‘𝑇)
78 eqid 2734 . . . 4 (-g𝑇) = (-g𝑇)
79 eqid 2734 . . . 4 (dist‘𝑇) = (dist‘𝑇)
8077, 78, 79isngp 24624 . . 3 (𝑇 ∈ NrmGrp ↔ (𝑇 ∈ Grp ∧ 𝑇 ∈ MetSp ∧ ((norm‘𝑇) ∘ (-g𝑇)) ⊆ (dist‘𝑇)))
8145, 65, 76, 80syl3anbrc 1342 . 2 ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))) → 𝑇 ∈ NrmGrp)
8243, 81impbida 801 1 (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962   × cxp 5686  cres 5690  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cr 11151  Basecbs 17244  +gcplusg 17297  distcds 17306  TopOpenctopn 17467  Grpcgrp 18963  -gcsg 18965  Metcmet 21367  MetOpencmopn 21371  MetSpcms 24343  normcnm 24604  NrmGrpcngp 24605   toNrmGrp ctng 24606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-tset 17316  df-ds 17319  df-rest 17468  df-topn 17469  df-0g 17487  df-topgen 17489  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-sbg 18968  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-xms 24345  df-ms 24346  df-nm 24610  df-ngp 24611  df-tng 24612
This theorem is referenced by:  tngngpd  24689  tngngp  24690  nrmtngnrm  24694  tngngpim  24695  tngnrg  24710
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