Theorem nmcvfval 29825

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 29818	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6882	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2761	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1542  ‘cfv 6535  2^nd c2nd 7961  norm_CVcnmcv 29808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3953  df-ss 3963  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-nmcv 29818

This theorem is referenced by:  nvop2  29826  nvop  29894  cnnvnm  29899  phop  30036  h2hnm  30194  hhssnm  30477