Theorem nmcvfval 30639

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 30632	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6921	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2768	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1537  ‘cfv 6573  2^nd c2nd 8029  norm_CVcnmcv 30622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-nmcv 30632

This theorem is referenced by:  nvop2  30640  nvop  30708  cnnvnm  30713  phop  30850  h2hnm  31008  hhssnm  31291