Theorem nmcvfval 30896

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 30889	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6880	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2792	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1567  ‘cfv 6534  2^nd c2nd 7981  norm_CVcnmcv 30879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-nmcv 30889

This theorem is referenced by:  nvop2  30897  nvop  30965  cnnvnm  30970  phop  31107  h2hnm  31265  hhssnm  31548