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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 2818 | . . . . . 6 ⊢ (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ (𝑋 × 𝑋)) | |
6 | 1, 2, 3, 4, 5 | isngp2 23133 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋)))) |
7 | 6 | simp3bi 1139 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
8 | 7 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
9 | 8 | oveqd 7162 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) |
10 | ngpgrp 23135 | . . . . . 6 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
11 | 4, 2 | grpsubf 18116 | . . . . . 6 ⊢ (𝐺 ∈ Grp → − :(𝑋 × 𝑋)⟶𝑋) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → − :(𝑋 × 𝑋)⟶𝑋) |
13 | 12 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → − :(𝑋 × 𝑋)⟶𝑋) |
14 | opelxpi 5585 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
15 | 14 | 3adant1 1122 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
16 | fvco3 6753 | . . . 4 ⊢ (( − :(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) | |
17 | 13, 15, 16 | syl2anc 584 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) |
18 | df-ov 7148 | . . 3 ⊢ (𝐴(𝑁 ∘ − )𝐵) = ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) | |
19 | df-ov 7148 | . . . 4 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
20 | 19 | fveq2i 6666 | . . 3 ⊢ (𝑁‘(𝐴 − 𝐵)) = (𝑁‘( − ‘〈𝐴, 𝐵〉)) |
21 | 17, 18, 20 | 3eqtr4g 2878 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝑁‘(𝐴 − 𝐵))) |
22 | ovres 7303 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
23 | 22 | 3adant1 1122 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
24 | 9, 21, 23 | 3eqtr3rd 2862 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 〈cop 4563 × cxp 5546 ↾ cres 5550 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 Grpcgrp 18041 -gcsg 18043 MetSpcms 22855 normcnm 23113 NrmGrpcngp 23114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 |
This theorem is referenced by: ngpdsr 23141 ngpds2 23142 ngprcan 23146 ngpinvds 23149 nmmtri 23158 nmrtri 23160 subgngp 23171 nrgdsdi 23201 nrgdsdir 23202 nlmdsdi 23217 nlmdsdir 23218 nrginvrcnlem 23227 nmods 23280 ncvspds 23692 ipcnlem2 23774 minveclem2 23956 minveclem3b 23958 minveclem4 23962 minveclem6 23964 |
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