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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 30543 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) |
6 | 1, 2, 5 | mp2an 690 | 1 ⊢ (𝑁‘𝐴) ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ‘cfv 6549 ℝcr 11139 NrmCVeccnv 30466 BaseSetcba 30468 normCVcnmcv 30472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-1st 7994 df-2nd 7995 df-vc 30441 df-nv 30474 df-va 30477 df-ba 30478 df-sm 30479 df-0v 30480 df-nmcv 30482 |
This theorem is referenced by: ip0i 30707 ip1ilem 30708 ipasslem10 30721 siilem1 30733 siii 30735 |
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