Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nrmtngnrm | Structured version Visualization version GIF version |
Description: The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
nrmtngdist.t | ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
Ref | Expression |
---|---|
nrmtngnrm | ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 22403 | . . 3 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
2 | nrmtngdist.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) | |
3 | eqid 2622 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | nrmtngdist 22461 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
5 | eqid 2622 | . . . . . 6 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
6 | eqid 2622 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
7 | eqid 2622 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
8 | eqid 2622 | . . . . . 6 ⊢ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) | |
9 | 5, 6, 7, 3, 8 | isngp2 22401 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))))) |
10 | 3, 8 | msmet 22262 | . . . . . 6 ⊢ (𝐺 ∈ MetSp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (Met‘(Base‘𝐺))) |
11 | 10 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))) → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (Met‘(Base‘𝐺))) |
12 | 9, 11 | sylbi 207 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (Met‘(Base‘𝐺))) |
13 | 4, 12 | eqeltrd 2701 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) ∈ (Met‘(Base‘𝐺))) |
14 | 3, 5 | nmf 22419 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝐺):(Base‘𝐺)⟶ℝ) |
15 | eqid 2622 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
16 | 2, 3, 15 | tngngp2 22456 | . . . 4 ⊢ ((norm‘𝐺):(Base‘𝐺)⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝐺))))) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝐺))))) |
18 | 1, 13, 17 | mpbir2and 957 | . 2 ⊢ (𝐺 ∈ NrmGrp → 𝑇 ∈ NrmGrp) |
19 | 1, 14 | jca 554 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝐺 ∈ Grp ∧ (norm‘𝐺):(Base‘𝐺)⟶ℝ)) |
20 | reex 10027 | . . . . 5 ⊢ ℝ ∈ V | |
21 | 2, 3, 20 | tngnm 22455 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ (norm‘𝐺):(Base‘𝐺)⟶ℝ) → (norm‘𝐺) = (norm‘𝑇)) |
22 | 19, 21 | syl 17 | . . 3 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝐺) = (norm‘𝑇)) |
23 | 22 | eqcomd 2628 | . 2 ⊢ (𝐺 ∈ NrmGrp → (norm‘𝑇) = (norm‘𝐺)) |
24 | 18, 23 | jca 554 | 1 ⊢ (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 Basecbs 15857 distcds 15950 Grpcgrp 17422 -gcsg 17424 Metcme 19732 MetSpcmt 22123 normcnm 22381 NrmGrpcngp 22382 toNrmGrp ctng 22383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-tset 15960 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-tng 22389 |
This theorem is referenced by: tngngpim 22463 |
Copyright terms: Public domain | W3C validator |