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Mirrors > Home > MPE Home > Th. List > ngpds | Structured version Visualization version GIF version |
Description: Value of the distance function in terms of the norm of a normed group. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpds.n | ⊢ 𝑁 = (norm‘𝐺) |
ngpds.x | ⊢ 𝑋 = (Base‘𝐺) |
ngpds.m | ⊢ − = (-g‘𝐺) |
ngpds.d | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
ngpds | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpds.n | . . . . . 6 ⊢ 𝑁 = (norm‘𝐺) | |
2 | ngpds.m | . . . . . 6 ⊢ − = (-g‘𝐺) | |
3 | ngpds.d | . . . . . 6 ⊢ 𝐷 = (dist‘𝐺) | |
4 | ngpds.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
5 | eqid 2622 | . . . . . 6 ⊢ (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ (𝑋 × 𝑋)) | |
6 | 1, 2, 3, 4, 5 | isngp2 22401 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋)))) |
7 | 6 | simp3bi 1078 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
8 | 7 | 3ad2ant1 1082 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
9 | 8 | oveqd 6667 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) |
10 | ngpgrp 22403 | . . . . . 6 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
11 | 4, 2 | grpsubf 17494 | . . . . . 6 ⊢ (𝐺 ∈ Grp → − :(𝑋 × 𝑋)⟶𝑋) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → − :(𝑋 × 𝑋)⟶𝑋) |
13 | 12 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → − :(𝑋 × 𝑋)⟶𝑋) |
14 | opelxpi 5148 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
15 | 14 | 3adant1 1079 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
16 | fvco3 6275 | . . . 4 ⊢ (( − :(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) | |
17 | 13, 15, 16 | syl2anc 693 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) |
18 | df-ov 6653 | . . 3 ⊢ (𝐴(𝑁 ∘ − )𝐵) = ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) | |
19 | df-ov 6653 | . . . 4 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
20 | 19 | fveq2i 6194 | . . 3 ⊢ (𝑁‘(𝐴 − 𝐵)) = (𝑁‘( − ‘〈𝐴, 𝐵〉)) |
21 | 17, 18, 20 | 3eqtr4g 2681 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝑁‘(𝐴 − 𝐵))) |
22 | ovres 6800 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
23 | 22 | 3adant1 1079 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
24 | 9, 21, 23 | 3eqtr3rd 2665 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 × cxp 5112 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 Grpcgrp 17422 -gcsg 17424 MetSpcmt 22123 normcnm 22381 NrmGrpcngp 22382 |
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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 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 |
This theorem is referenced by: ngpdsr 22409 ngpds2 22410 ngprcan 22414 ngpinvds 22417 nmmtri 22426 nmrtri 22428 subgngp 22439 nrgdsdi 22469 nrgdsdir 22470 nlmdsdi 22485 nlmdsdir 22486 nrginvrcnlem 22495 nmods 22548 ncvspds 22961 ipcnlem2 23043 minveclem2 23197 minveclem3b 23199 minveclem4 23203 minveclem6 23205 |
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