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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 6669 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | imsmet.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | fveq2 6669 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
5 | 3, 4 | syl5eq 2868 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑋 = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
6 | 5 | fveq2d 6673 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (Met‘𝑋) = (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
7 | 2, 6 | eleq12d 2907 | . . 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 2821 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
9 | eqid 2821 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
10 | eqid 2821 | . . . 4 ⊢ (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) = (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
11 | eqid 2821 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
12 | eqid 2821 | . . . 4 ⊢ (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
13 | eqid 2821 | . . . 4 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
14 | eqid 2821 | . . . 4 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
15 | elimnvu 28460 | . . . 4 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
16 | 8, 9, 10, 11, 12, 13, 14, 15 | imsmetlem 28466 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
17 | 7, 16 | dedth 4522 | . 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (Met‘𝑋)) |
18 | 1, 17 | eqeltrid 2917 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ifcif 4466 〈cop 4572 ‘cfv 6354 + caddc 10539 · cmul 10541 abscabs 14592 Metcmet 20530 invcgn 28267 NrmCVeccnv 28360 +𝑣 cpv 28361 BaseSetcba 28362 ·𝑠OLD cns 28363 0veccn0v 28364 normCVcnmcv 28366 IndMetcims 28367 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-met 20538 df-grpo 28269 df-gid 28270 df-ginv 28271 df-gdiv 28272 df-ablo 28321 df-vc 28335 df-nv 28368 df-va 28371 df-ba 28372 df-sm 28373 df-0v 28374 df-vs 28375 df-nmcv 28376 df-ims 28377 |
This theorem is referenced by: imsxmet 28468 vacn 28470 nmcvcn 28471 smcnlem 28473 blocni 28581 minvecolem2 28651 minvecolem3 28652 minvecolem4a 28653 minvecolem4 28656 minvecolem7 28659 hhmet 28950 hhssmet 29052 |
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