MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tngnm Structured version   Visualization version   GIF version

Theorem tngnm 22365
Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngnm.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngnm.x 𝑋 = (Base‘𝐺)
tngnm.a 𝐴 ∈ V
Assertion
Ref Expression
tngnm ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))

Proof of Theorem tngnm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . 3 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁:𝑋𝐴)
21feqmptd 6206 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (𝑥𝑋 ↦ (𝑁𝑥)))
3 tngnm.x . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2621 . . . . . . . 8 (-g𝐺) = (-g𝐺)
53, 4grpsubf 17415 . . . . . . 7 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
65ad2antrr 761 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
7 simpr 477 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
8 eqid 2621 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
93, 8grpidcl 17371 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
109ad2antrr 761 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑋)
11 opelxpi 5108 . . . . . . 7 ((𝑥𝑋 ∧ (0g𝐺) ∈ 𝑋) → ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋))
127, 10, 11syl2anc 692 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋))
13 fvco3 6232 . . . . . 6 (((-g𝐺):(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)))
146, 12, 13syl2anc 692 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)))
15 df-ov 6607 . . . . 5 (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩)
16 df-ov 6607 . . . . . 6 (𝑥(-g𝐺)(0g𝐺)) = ((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)
1716fveq2i 6151 . . . . 5 (𝑁‘(𝑥(-g𝐺)(0g𝐺))) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩))
1814, 15, 173eqtr4g 2680 . . . 4 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = (𝑁‘(𝑥(-g𝐺)(0g𝐺))))
193, 8, 4grpsubid1 17421 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑥(-g𝐺)(0g𝐺)) = 𝑥)
2019adantlr 750 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑥(-g𝐺)(0g𝐺)) = 𝑥)
2120fveq2d 6152 . . . 4 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑁‘(𝑥(-g𝐺)(0g𝐺))) = (𝑁𝑥))
2218, 21eqtr2d 2656 . . 3 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑁𝑥) = (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)))
2322mpteq2dva 4704 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑁𝑥)) = (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))))
24 fvex 6158 . . . . . . . 8 (Base‘𝐺) ∈ V
253, 24eqeltri 2694 . . . . . . 7 𝑋 ∈ V
26 tngnm.a . . . . . . 7 𝐴 ∈ V
27 fex2 7068 . . . . . . 7 ((𝑁:𝑋𝐴𝑋 ∈ V ∧ 𝐴 ∈ V) → 𝑁 ∈ V)
2825, 26, 27mp3an23 1413 . . . . . 6 (𝑁:𝑋𝐴𝑁 ∈ V)
2928adantl 482 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 ∈ V)
30 tngnm.t . . . . . 6 𝑇 = (𝐺 toNrmGrp 𝑁)
3130, 3tngbas 22355 . . . . 5 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
3229, 31syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑋 = (Base‘𝑇))
3330, 4tngds 22362 . . . . . 6 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3429, 33syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
35 eqidd 2622 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑥 = 𝑥)
3630, 8tng0 22357 . . . . . 6 (𝑁 ∈ V → (0g𝐺) = (0g𝑇))
3729, 36syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (0g𝐺) = (0g𝑇))
3834, 35, 37oveq123d 6625 . . . 4 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = (𝑥(dist‘𝑇)(0g𝑇)))
3932, 38mpteq12dv 4693 . . 3 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g𝑇))))
40 eqid 2621 . . . 4 (norm‘𝑇) = (norm‘𝑇)
41 eqid 2621 . . . 4 (Base‘𝑇) = (Base‘𝑇)
42 eqid 2621 . . . 4 (0g𝑇) = (0g𝑇)
43 eqid 2621 . . . 4 (dist‘𝑇) = (dist‘𝑇)
4440, 41, 42, 43nmfval 22303 . . 3 (norm‘𝑇) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g𝑇)))
4539, 44syl6eqr 2673 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))) = (norm‘𝑇))
462, 23, 453eqtrd 2659 1 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  cmpt 4673   × cxp 5072  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  0gc0g 16021  Grpcgrp 17343  -gcsg 17345  normcnm 22291   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
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-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  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-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-plusg 15875  df-tset 15881  df-ds 15885  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-nm 22297  df-tng 22299
This theorem is referenced by:  tngngp2  22366  tngngp  22368  tngngp3  22370  nrmtngnrm  22372  tngnrg  22388  tchnmfval  22935  tchcph  22944
  Copyright terms: Public domain W3C validator