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Mirrors > Home > MPE Home > Th. List > nmcnc | Structured version Visualization version GIF version |
Description: The norm of a normed complex vector space is a continuous function to ℂ. (For ℝ, see nmcvcn 28122.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmcnc.1 | ⊢ 𝑁 = (normCV‘𝑈) |
nmcnc.2 | ⊢ 𝐶 = (IndMet‘𝑈) |
nmcnc.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
nmcnc.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
nmcnc | ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmcnc.k | . . . 4 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtop 22995 | . . 3 ⊢ 𝐾 ∈ Top |
3 | cnrest2r 21499 | . . 3 ⊢ (𝐾 ∈ Top → (𝐽 Cn (𝐾 ↾t ℝ)) ⊆ (𝐽 Cn 𝐾)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (𝐽 Cn (𝐾 ↾t ℝ)) ⊆ (𝐽 Cn 𝐾) |
5 | nmcnc.1 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
6 | nmcnc.2 | . . 3 ⊢ 𝐶 = (IndMet‘𝑈) | |
7 | nmcnc.j | . . 3 ⊢ 𝐽 = (MetOpen‘𝐶) | |
8 | 1 | tgioo2 23014 | . . . 4 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
9 | 8 | eqcomi 2786 | . . 3 ⊢ (𝐾 ↾t ℝ) = (topGen‘ran (,)) |
10 | 5, 6, 7, 9 | nmcvcn 28122 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn (𝐾 ↾t ℝ))) |
11 | 4, 10 | sseldi 3818 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 ran crn 5356 ‘cfv 6135 (class class class)co 6922 ℝcr 10271 (,)cioo 12487 ↾t crest 16467 TopOpenctopn 16468 topGenctg 16484 MetOpencmopn 20132 ℂfldccnfld 20142 Topctop 21105 Cn ccn 21436 NrmCVeccnv 28011 normCVcnmcv 28017 IndMetcims 28018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-topgen 16490 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-xms 22533 df-ms 22534 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 |
This theorem is referenced by: dipcn 28147 |
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