Theorem nmcvfval 26846

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 26839	. . 3 ⊢ normCV = 2nd
3	2	fveq1i 6104	. 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈)
4	1, 3	eqtri 2632	1 ⊢ 𝑁 = (2nd ‘𝑈)

Syntax hints:   = wceq 1475  ‘cfv 5804  2^nd c2nd 7058  norm_CVcnmcv 26829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-nmcv 26839

This theorem is referenced by:  nvop2  26847  nvop  26915  cnnvnm  26920  phop  27057  phpar  27063  h2hnm  27217  hhssnm  27500