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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 2821 | . . . . . 6 ⊢ (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ (𝑋 × 𝑋)) | |
6 | 1, 2, 3, 4, 5 | isngp2 23200 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋)))) |
7 | 6 | simp3bi 1143 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
8 | 7 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
9 | 8 | oveqd 7167 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) |
10 | ngpgrp 23202 | . . . . . 6 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
11 | 4, 2 | grpsubf 18172 | . . . . . 6 ⊢ (𝐺 ∈ Grp → − :(𝑋 × 𝑋)⟶𝑋) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → − :(𝑋 × 𝑋)⟶𝑋) |
13 | 12 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → − :(𝑋 × 𝑋)⟶𝑋) |
14 | opelxpi 5586 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
15 | 14 | 3adant1 1126 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
16 | fvco3 6754 | . . . 4 ⊢ (( − :(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) | |
17 | 13, 15, 16 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) = (𝑁‘( − ‘〈𝐴, 𝐵〉))) |
18 | df-ov 7153 | . . 3 ⊢ (𝐴(𝑁 ∘ − )𝐵) = ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) | |
19 | df-ov 7153 | . . . 4 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
20 | 19 | fveq2i 6667 | . . 3 ⊢ (𝑁‘(𝐴 − 𝐵)) = (𝑁‘( − ‘〈𝐴, 𝐵〉)) |
21 | 17, 18, 20 | 3eqtr4g 2881 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝑁‘(𝐴 − 𝐵))) |
22 | ovres 7308 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
23 | 22 | 3adant1 1126 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
24 | 9, 21, 23 | 3eqtr3rd 2865 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 〈cop 4566 × cxp 5547 ↾ cres 5551 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 distcds 16568 Grpcgrp 18097 -gcsg 18099 MetSpcms 22922 normcnm 23180 NrmGrpcngp 23181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 |
This theorem is referenced by: ngpdsr 23208 ngpds2 23209 ngprcan 23213 ngpinvds 23216 nmmtri 23225 nmrtri 23227 subgngp 23238 nrgdsdi 23268 nrgdsdir 23269 nlmdsdi 23284 nlmdsdir 23285 nrginvrcnlem 23294 nmods 23347 ncvspds 23759 ipcnlem2 23841 minveclem2 24023 minveclem3b 24025 minveclem4 24029 minveclem6 24031 |
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