Theorem nmcvfval 30128

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 30121	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6892	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2759	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1540  ‘cfv 6543  2^nd c2nd 7978  norm_CVcnmcv 30111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-nmcv 30121

This theorem is referenced by:  nvop2  30129  nvop  30197  cnnvnm  30202  phop  30339  h2hnm  30497  hhssnm  30780