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Theorem nvcli 30544
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1 𝑋 = (BaseSet‘𝑈)
nvf.6 𝑁 = (normCV𝑈)
nvcli.9 𝑈 ∈ NrmCVec
nvcli.7 𝐴𝑋
Assertion
Ref Expression
nvcli (𝑁𝐴) ∈ ℝ

Proof of Theorem nvcli
StepHypRef Expression
1 nvcli.9 . 2 𝑈 ∈ NrmCVec
2 nvcli.7 . 2 𝐴𝑋
3 nvf.1 . . 3 𝑋 = (BaseSet‘𝑈)
4 nvf.6 . . 3 𝑁 = (normCV𝑈)
53, 4nvcl 30543 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) ∈ ℝ)
61, 2, 5mp2an 690 1 (𝑁𝐴) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  cfv 6549  cr 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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