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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 2737 | . . . . . 6 ⊢ (𝐷 ↾ (𝑋 × 𝑋)) = (𝐷 ↾ (𝑋 × 𝑋)) | |
6 | 1, 2, 3, 4, 5 | isngp2 23904 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋)))) |
7 | 6 | simp3bi 1147 | . . . 4 ⊢ (𝐺 ∈ NrmGrp → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
8 | 7 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁 ∘ − ) = (𝐷 ↾ (𝑋 × 𝑋))) |
9 | 8 | oveqd 7368 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵)) |
10 | ngpgrp 23906 | . . . . . 6 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
11 | 4, 2 | grpsubf 18784 | . . . . . 6 ⊢ (𝐺 ∈ Grp → − :(𝑋 × 𝑋)⟶𝑋) |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → − :(𝑋 × 𝑋)⟶𝑋) |
13 | 12 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → − :(𝑋 × 𝑋)⟶𝑋) |
14 | opelxpi 5668 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
15 | 14 | 3adant1 1130 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) |
16 | fvco3 6937 | . . . 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 7354 | . . 3 ⊢ (𝐴(𝑁 ∘ − )𝐵) = ((𝑁 ∘ − )‘〈𝐴, 𝐵〉) | |
19 | df-ov 7354 | . . . 4 ⊢ (𝐴 − 𝐵) = ( − ‘〈𝐴, 𝐵〉) | |
20 | 19 | fveq2i 6842 | . . 3 ⊢ (𝑁‘(𝐴 − 𝐵)) = (𝑁‘( − ‘〈𝐴, 𝐵〉)) |
21 | 17, 18, 20 | 3eqtr4g 2802 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑁 ∘ − )𝐵) = (𝑁‘(𝐴 − 𝐵))) |
22 | ovres 7514 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
23 | 22 | 3adant1 1130 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
24 | 9, 21, 23 | 3eqtr3rd 2786 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 〈cop 4590 × cxp 5629 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 distcds 17101 Grpcgrp 18707 -gcsg 18709 MetSpcms 23622 normcnm 23883 NrmGrpcngp 23884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-0g 17282 df-topgen 17284 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-psmet 20740 df-xmet 20741 df-met 20742 df-bl 20743 df-mopn 20744 df-top 22194 df-topon 22211 df-topsp 22233 df-bases 22247 df-xms 23624 df-ms 23625 df-nm 23889 df-ngp 23890 |
This theorem is referenced by: ngpdsr 23912 ngpds2 23913 ngprcan 23917 ngpinvds 23920 nmmtri 23929 nmrtri 23931 subgngp 23942 nrgdsdi 23980 nrgdsdir 23981 nlmdsdi 23996 nlmdsdir 23997 nrginvrcnlem 24006 nmods 24059 ncvspds 24476 ipcnlem2 24559 minveclem2 24741 minveclem3b 24743 minveclem4 24747 minveclem6 24749 |
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