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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 29488 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) |
4 | 3 | ffvelcdmda 7031 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 ℝcr 11046 NrmCVeccnv 29412 BaseSetcba 29414 normCVcnmcv 29418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-1st 7917 df-2nd 7918 df-vc 29387 df-nv 29420 df-va 29423 df-ba 29424 df-sm 29425 df-0v 29426 df-nmcv 29428 |
This theorem is referenced by: nvcli 29490 nvm1 29493 nvpi 29495 nvz0 29496 nvmtri 29499 nvabs 29500 nvge0 29501 nvgt0 29502 nv1 29503 nmcvcn 29523 smcnlem 29525 ipval2lem2 29532 4ipval2 29536 ipidsq 29538 ipnm 29539 ipz 29547 nmosetre 29592 nmooge0 29595 nmoub3i 29601 nmounbi 29604 nmlno0lem 29621 nmblolbii 29627 blocnilem 29632 ipblnfi 29683 ubthlem1 29698 ubthlem2 29699 ubthlem3 29700 minvecolem1 29702 minvecolem2 29703 minvecolem4 29708 minvecolem5 29709 minvecolem6 29710 hlipgt0 29742 htthlem 29745 |
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