Theorem nmcvfval 30678

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 30671	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6841	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2759	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1542  ‘cfv 6498  2^nd c2nd 7941  norm_CVcnmcv 30661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-nmcv 30671

This theorem is referenced by:  nvop2  30679  nvop  30747  cnnvnm  30752  phop  30889  h2hnm  31047  hhssnm  31330