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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 28431 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) |
4 | 3 | ffvelrnda 6846 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 ℝcr 10530 NrmCVeccnv 28355 BaseSetcba 28357 normCVcnmcv 28361 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-1st 7683 df-2nd 7684 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 |
This theorem is referenced by: nvcli 28433 nvm1 28436 nvpi 28438 nvz0 28439 nvmtri 28442 nvabs 28443 nvge0 28444 nvgt0 28445 nv1 28446 nmcvcn 28466 smcnlem 28468 ipval2lem2 28475 4ipval2 28479 ipidsq 28481 ipnm 28482 ipz 28490 nmosetre 28535 nmooge0 28538 nmoub3i 28544 nmounbi 28547 nmlno0lem 28564 nmblolbii 28570 blocnilem 28575 ipblnfi 28626 ubthlem1 28641 ubthlem2 28642 ubthlem3 28643 minvecolem1 28645 minvecolem2 28646 minvecolem4 28651 minvecolem5 28652 minvecolem6 28653 hlipgt0 28685 htthlem 28688 |
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