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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 29936.) (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 24292 | . . 3 ⊢ 𝐾 ∈ Top |
3 | cnrest2r 22783 | . . 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 24311 | . . . 4 ⊢ (topGen‘ran (,)) = (𝐾 ↾t ℝ) |
9 | 8 | eqcomi 2742 | . . 3 ⊢ (𝐾 ↾t ℝ) = (topGen‘ran (,)) |
10 | 5, 6, 7, 9 | nmcvcn 29936 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn (𝐾 ↾t ℝ))) |
11 | 4, 10 | sselid 3980 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 ran crn 5677 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 (,)cioo 13321 ↾t crest 17363 TopOpenctopn 17364 topGenctg 17380 MetOpencmopn 20927 ℂfldccnfld 20937 Topctop 22387 Cn ccn 22720 NrmCVeccnv 29825 normCVcnmcv 29831 IndMetcims 29832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-fz 13482 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-rest 17365 df-topn 17366 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cn 22723 df-cnp 22724 df-xms 23818 df-ms 23819 df-grpo 29734 df-gid 29735 df-ginv 29736 df-gdiv 29737 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-vs 29840 df-nmcv 29841 df-ims 29842 |
This theorem is referenced by: dipcn 29961 |
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