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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 30679 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) |
| 4 | 3 | ffvelcdmda 7104 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℝcr 11154 NrmCVeccnv 30603 BaseSetcba 30605 normCVcnmcv 30609 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 |
| This theorem is referenced by: nvcli 30681 nvm1 30684 nvpi 30686 nvz0 30687 nvmtri 30690 nvabs 30691 nvge0 30692 nvgt0 30693 nv1 30694 nmcvcn 30714 smcnlem 30716 ipval2lem2 30723 4ipval2 30727 ipidsq 30729 ipnm 30730 ipz 30738 nmosetre 30783 nmooge0 30786 nmoub3i 30792 nmounbi 30795 nmlno0lem 30812 nmblolbii 30818 blocnilem 30823 ipblnfi 30874 ubthlem1 30889 ubthlem2 30890 ubthlem3 30891 minvecolem1 30893 minvecolem2 30894 minvecolem4 30899 minvecolem5 30900 minvecolem6 30901 hlipgt0 30933 htthlem 30936 |
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