Theorem nmcvfval 28688

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 28681	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6718	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2765	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1543  ‘cfv 6380  2^nd c2nd 7760  norm_CVcnmcv 28671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-nmcv 28681

This theorem is referenced by:  nvop2  28689  nvop  28757  cnnvnm  28762  phop  28899  h2hnm  29057  hhssnm  29340