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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 26899 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) |
4 | 3 | ffvelrnda 6267 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 ℝcr 9814 NrmCVeccnv 26823 BaseSetcba 26825 normCVcnmcv 26829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 |
This theorem is referenced by: nvcli 26901 nvm1 26904 nvpi 26906 nvz0 26907 nvmtri 26910 nvabs 26911 nvge0 26912 nvgt0 26913 nv1 26914 nmcvcn 26934 smcnlem 26936 ipval2lem2 26943 4ipval2 26947 ipidsq 26949 ipnm 26950 ipz 26958 nmosetre 27003 nmooge0 27006 nmoub3i 27012 nmounbi 27015 nmlno0lem 27032 nmblolbii 27038 blocnilem 27043 ipblnfi 27095 ubthlem1 27110 ubthlem2 27111 ubthlem3 27112 minvecolem1 27114 minvecolem2 27115 minvecolem4 27120 minvecolem5 27121 minvecolem6 27122 hlipgt0 27154 htthlem 27158 |
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