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Mirrors > Home > MPE Home > Th. List > imsmet | Structured version Visualization version GIF version |
Description: The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsmet.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
imsmet.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsmet | ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsmet.8 | . 2 ⊢ 𝐷 = (IndMet‘𝑈) | |
2 | fveq2 6663 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | imsmet.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | fveq2 6663 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
5 | 3, 4 | syl5eq 2865 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑋 = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
6 | 5 | fveq2d 6667 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (Met‘𝑋) = (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
7 | 2, 6 | eleq12d 2904 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → ((IndMet‘𝑈) ∈ (Met‘𝑋) ↔ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))))) |
8 | eqid 2818 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
9 | eqid 2818 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
10 | eqid 2818 | . . . 4 ⊢ (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) = (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
11 | eqid 2818 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
12 | eqid 2818 | . . . 4 ⊢ (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
13 | eqid 2818 | . . . 4 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
14 | eqid 2818 | . . . 4 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
15 | elimnvu 28388 | . . . 4 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
16 | 8, 9, 10, 11, 12, 13, 14, 15 | imsmetlem 28394 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
17 | 7, 16 | dedth 4519 | . 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (Met‘𝑋)) |
18 | 1, 17 | eqeltrid 2914 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ifcif 4463 〈cop 4563 ‘cfv 6348 + caddc 10528 · cmul 10530 abscabs 14581 Metcmet 20459 invcgn 28195 NrmCVeccnv 28288 +𝑣 cpv 28289 BaseSetcba 28290 ·𝑠OLD cns 28291 0veccn0v 28292 normCVcnmcv 28294 IndMetcims 28295 |
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-rep 5181 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 ax-addf 10604 ax-mulf 10605 |
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-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-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-met 20467 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 |
This theorem is referenced by: imsxmet 28396 vacn 28398 nmcvcn 28399 smcnlem 28401 blocni 28509 minvecolem2 28579 minvecolem3 28580 minvecolem4a 28581 minvecolem4 28584 minvecolem7 28587 hhmet 28878 hhssmet 28980 |
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