Theorem nmcvfval 30627

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 30620	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6906	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2764	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1539  ‘cfv 6560  2^nd c2nd 8014  norm_CVcnmcv 30610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-nmcv 30620

This theorem is referenced by:  nvop2  30628  nvop  30696  cnnvnm  30701  phop  30838  h2hnm  30996  hhssnm  31279