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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 6884 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) | |
3 | imsmet.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | fveq2 6884 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) | |
5 | 3, 4 | eqtrid 2778 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → 𝑋 = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) |
6 | 5 | fveq2d 6888 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) → (Met‘𝑋) = (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)))) |
7 | 2, 6 | eleq12d 2821 | . . 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 2726 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
9 | eqid 2726 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
10 | eqid 2726 | . . . 4 ⊢ (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) = (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) | |
11 | eqid 2726 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
12 | eqid 2726 | . . . 4 ⊢ (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
13 | eqid 2726 | . . . 4 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
14 | eqid 2726 | . . . 4 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) | |
15 | elimnvu 30442 | . . . 4 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ NrmCVec | |
16 | 8, 9, 10, 11, 12, 13, 14, 15 | imsmetlem 30448 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩))) |
17 | 7, 16 | dedth 4581 | . 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (Met‘𝑋)) |
18 | 1, 17 | eqeltrid 2831 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ifcif 4523 ⟨cop 4629 ‘cfv 6536 + caddc 11112 · cmul 11114 abscabs 15185 Metcmet 21222 invcgn 30249 NrmCVeccnv 30342 +𝑣 cpv 30343 BaseSetcba 30344 ·𝑠OLD cns 30345 0veccn0v 30346 normCVcnmcv 30348 IndMetcims 30349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-met 21230 df-grpo 30251 df-gid 30252 df-ginv 30253 df-gdiv 30254 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-vs 30357 df-nmcv 30358 df-ims 30359 |
This theorem is referenced by: imsxmet 30450 vacn 30452 nmcvcn 30453 smcnlem 30455 blocni 30563 minvecolem2 30633 minvecolem3 30634 minvecolem4a 30635 minvecolem4 30638 minvecolem7 30641 hhmet 30932 hhssmet 31034 |
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