Theorem nmcvfval 30636

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 30629	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6908	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2763	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1537  ‘cfv 6563  2^nd c2nd 8012  norm_CVcnmcv 30619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-nmcv 30629

This theorem is referenced by:  nvop2  30637  nvop  30705  cnnvnm  30710  phop  30847  h2hnm  31005  hhssnm  31288