Theorem nmcvfval 30693

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 30686	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6835	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2760	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1542  ‘cfv 6492  2^nd c2nd 7934  norm_CVcnmcv 30676

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-ss 3907  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-nmcv 30686

This theorem is referenced by:  nvop2  30694  nvop  30762  cnnvnm  30767  phop  30904  h2hnm  31062  hhssnm  31345