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Theorem nvcli 29701
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 = (BaseSetβ€˜π‘ˆ)
nvf.6 𝑁 = (normCVβ€˜π‘ˆ)
nvcli.9 π‘ˆ ∈ NrmCVec
nvcli.7 𝐴 ∈ 𝑋
Assertion
Ref Expression
nvcli (π‘β€˜π΄) ∈ ℝ

Proof of Theorem nvcli
StepHypRef Expression
1 nvcli.9 . 2 π‘ˆ ∈ NrmCVec
2 nvcli.7 . 2 𝐴 ∈ 𝑋
3 nvf.1 . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
4 nvf.6 . . 3 𝑁 = (normCVβ€˜π‘ˆ)
53, 4nvcl 29700 . 2 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ ℝ)
61, 2, 5mp2an 690 1 (π‘β€˜π΄) ∈ ℝ
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   ∈ wcel 2106  β€˜cfv 6516  β„cr 11074  NrmCVeccnv 29623  BaseSetcba 29625  normCVcnmcv 29629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-1st 7941  df-2nd 7942  df-vc 29598  df-nv 29631  df-va 29634  df-ba 29635  df-sm 29636  df-0v 29637  df-nmcv 29639
This theorem is referenced by:  ip0i  29864  ip1ilem  29865  ipasslem10  29878  siilem1  29890  siii  29892
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