Theorem nmcvfval 30593

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 30586	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6882	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2759	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1540  ‘cfv 6536  2^nd c2nd 7992  norm_CVcnmcv 30576

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-ss 3948  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-nmcv 30586

This theorem is referenced by:  nvop2  30594  nvop  30662  cnnvnm  30667  phop  30804  h2hnm  30962  hhssnm  31245