Theorem nmcvfval 30696

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 30689	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6828	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2762	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1547  ‘cfv 6485  2^nd c2nd 7930  norm_CVcnmcv 30679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-ss 3900  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-nmcv 30689

This theorem is referenced by:  nvop2  30697  nvop  30765  cnnvnm  30770  phop  30907  h2hnm  31065  hhssnm  31348