Theorem nmcvfval 30754

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 30747	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6862	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2784	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1559  ‘cfv 6515  2^nd c2nd 7963  norm_CVcnmcv 30737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3921  df-uni 4865  df-br 5100  df-iota 6471  df-fv 6523  df-nmcv 30747

This theorem is referenced by:  nvop2  30755  nvop  30823  cnnvnm  30828  phop  30965  h2hnm  31123  hhssnm  31406