Theorem nmcvfval 30959

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 30952	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6882	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2786	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1570  ‘cfv 6536  2^nd c2nd 7981  norm_CVcnmcv 30942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-nmcv 30952

This theorem is referenced by:  nvop2  30960  nvop  31028  cnnvnm  31033  phop  31170  h2hnm  31328  hhssnm  31611