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Mirrors > Home > MPE Home > Th. List > nvcl | Structured version Visualization version GIF version |
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvf.6 | ⊢ 𝑁 = (normCV‘𝑈) |
Ref | Expression |
---|---|
nvcl | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | nvf.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
3 | 1, 2 | nvf 28040 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) |
4 | 3 | ffvelrnda 6585 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ‘cfv 6101 ℝcr 10223 NrmCVeccnv 27964 BaseSetcba 27966 normCVcnmcv 27970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-1st 7401 df-2nd 7402 df-vc 27939 df-nv 27972 df-va 27975 df-ba 27976 df-sm 27977 df-0v 27978 df-nmcv 27980 |
This theorem is referenced by: nvcli 28042 nvm1 28045 nvpi 28047 nvz0 28048 nvmtri 28051 nvabs 28052 nvge0 28053 nvgt0 28054 nv1 28055 nmcvcn 28075 smcnlem 28077 ipval2lem2 28084 4ipval2 28088 ipidsq 28090 ipnm 28091 ipz 28099 nmosetre 28144 nmooge0 28147 nmoub3i 28153 nmounbi 28156 nmlno0lem 28173 nmblolbii 28179 blocnilem 28184 ipblnfi 28236 ubthlem1 28251 ubthlem2 28252 ubthlem3 28253 minvecolem1 28255 minvecolem2 28256 minvecolem4 28261 minvecolem5 28262 minvecolem6 28263 hlipgt0 28295 htthlem 28299 |
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