Theorem nmcvfval 29612

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 29605	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6848	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2759	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1541  ‘cfv 6501  2^nd c2nd 7925  norm_CVcnmcv 29595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-nmcv 29605

This theorem is referenced by:  nvop2  29613  nvop  29681  cnnvnm  29686  phop  29823  h2hnm  29981  hhssnm  30264