Theorem nmcvfval 28384

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 28377	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6671	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2844	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1537  ‘cfv 6355  2^nd c2nd 7688  norm_CVcnmcv 28367

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3943  df-ss 3952  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-nmcv 28377

This theorem is referenced by:  nvop2  28385  nvop  28453  cnnvnm  28458  phop  28595  h2hnm  28753  hhssnm  29036