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Mirrors > Home > MPE Home > Th. List > nrmtngdist | Structured version Visualization version GIF version |
Description: The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
nrmtngdist.t | ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) |
nrmtngdist.x | ⊢ 𝑋 = (Base‘𝐺) |
Ref | Expression |
---|---|
nrmtngdist | ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6239 | . . 3 ⊢ (norm‘𝐺) ∈ V | |
2 | nrmtngdist.t | . . . 4 ⊢ 𝑇 = (𝐺 toNrmGrp (norm‘𝐺)) | |
3 | eqid 2651 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
4 | 2, 3 | tngds 22499 | . . 3 ⊢ ((norm‘𝐺) ∈ V → ((norm‘𝐺) ∘ (-g‘𝐺)) = (dist‘𝑇)) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ((norm‘𝐺) ∘ (-g‘𝐺)) = (dist‘𝑇) |
6 | eqid 2651 | . . . 4 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
7 | eqid 2651 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
8 | nrmtngdist.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
9 | eqid 2651 | . . . 4 ⊢ ((dist‘𝐺) ↾ (𝑋 × 𝑋)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) | |
10 | 6, 3, 7, 8, 9 | isngp2 22448 | . . 3 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)))) |
11 | 10 | simp3bi 1098 | . 2 ⊢ (𝐺 ∈ NrmGrp → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
12 | 5, 11 | syl5eqr 2699 | 1 ⊢ (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 × cxp 5141 ↾ cres 5145 ∘ ccom 5147 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 distcds 15997 Grpcgrp 17469 -gcsg 17471 MetSpcmt 22170 normcnm 22428 NrmGrpcngp 22429 toNrmGrp ctng 22430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ndx 15907 df-slot 15908 df-sets 15911 df-tset 16007 df-ds 16011 df-0g 16149 df-topgen 16151 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-xms 22172 df-ms 22173 df-nm 22434 df-ngp 22435 df-tng 22436 |
This theorem is referenced by: nrmtngnrm 22509 |
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