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Mirrors > Home > MPE Home > Th. List > nvcli | Structured version Visualization version GIF version |
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
nvf.6 | ⊢ 𝑁 = (normCV‘𝑈) |
nvcli.9 | ⊢ 𝑈 ∈ NrmCVec |
nvcli.7 | ⊢ 𝐴 ∈ 𝑋 |
Ref | Expression |
---|---|
nvcli | ⊢ (𝑁‘𝐴) ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvcli.9 | . 2 ⊢ 𝑈 ∈ NrmCVec | |
2 | nvcli.7 | . 2 ⊢ 𝐴 ∈ 𝑋 | |
3 | nvf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | nvf.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
5 | 3, 4 | nvcl 30690 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (𝑁‘𝐴) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ‘cfv 6563 ℝcr 11152 NrmCVeccnv 30613 BaseSetcba 30615 normCVcnmcv 30619 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-1st 8013 df-2nd 8014 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-nmcv 30629 |
This theorem is referenced by: ip0i 30854 ip1ilem 30855 ipasslem10 30868 siilem1 30880 siii 30882 |
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