Theorem nmcvfval 30579

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 30572	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6818	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2754	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1541  ‘cfv 6476  2^nd c2nd 7915  norm_CVcnmcv 30562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3914  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-nmcv 30572

This theorem is referenced by:  nvop2  30580  nvop  30648  cnnvnm  30653  phop  30790  h2hnm  30948  hhssnm  31231