Theorem nmcvfval 30127

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 30120	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6891	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2758	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1539  ‘cfv 6542  2^nd c2nd 7976  norm_CVcnmcv 30110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-nmcv 30120

This theorem is referenced by:  nvop2  30128  nvop  30196  cnnvnm  30201  phop  30338  h2hnm  30496  hhssnm  30779