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Theorem nvcl 29700
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nvf.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
nvcl ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ ℝ)

Proof of Theorem nvcl
StepHypRef Expression
1 nvf.1 . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
2 nvf.6 . . 3 𝑁 = (normCVβ€˜π‘ˆ)
31, 2nvf 29699 . 2 (π‘ˆ ∈ NrmCVec β†’ 𝑁:π‘‹βŸΆβ„)
43ffvelcdmda 7055 1 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = 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:  nvcli  29701  nvm1  29704  nvpi  29706  nvz0  29707  nvmtri  29710  nvabs  29711  nvge0  29712  nvgt0  29713  nv1  29714  nmcvcn  29734  smcnlem  29736  ipval2lem2  29743  4ipval2  29747  ipidsq  29749  ipnm  29750  ipz  29758  nmosetre  29803  nmooge0  29806  nmoub3i  29812  nmounbi  29815  nmlno0lem  29832  nmblolbii  29838  blocnilem  29843  ipblnfi  29894  ubthlem1  29909  ubthlem2  29910  ubthlem3  29911  minvecolem1  29913  minvecolem2  29914  minvecolem4  29919  minvecolem5  29920  minvecolem6  29921  hlipgt0  29953  htthlem  29956
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