Theorem nmcvfval 28948

Description: Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nmfval.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
nmcvfval	⊢ 𝑁 = (2nd ‘𝑈)

Proof of Theorem nmcvfval

Step	Hyp	Ref	Expression
1		nmfval.6	. 2 ⊢ 𝑁 = (normCV‘𝑈)
2		df-nmcv 28941	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6769	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2767	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1541  ‘cfv 6430  2^nd c2nd 7816  norm_CVcnmcv 28931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-in 3898  df-ss 3908  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-nmcv 28941

This theorem is referenced by:  nvop2  28949  nvop  29017  cnnvnm  29022  phop  29159  h2hnm  29317  hhssnm  29600