Theorem nmfval 8222

Description: Value of the norm function in a normed complex vector space.

Hypothesis

Ref	Expression
nmfval.6	|- N = (norm` U)

Assertion

Ref	Expression
nmfval	|- N = (2nd` U)

Proof of Theorem nmfval

Step	Hyp	Ref	Expression
1		nmfval.6	. 2 |- N = (norm` U)
2		df-nm 8215	. . 3 |- norm = 2nd
3	2	fveq1i 3731	. 2 |- (norm` U) = (2nd` U)
4	1, 3	eqtr 1498	1 |- N = (2nd` U)

Syntax hints:   = wceq 958  ` cfv 3188  2ndc2nd 4084  normcnm 8205

This theorem is referenced by:  nvop2 8223  nvi 8229  nvvc 8230  nvop 8301  cnnvnm 8308  abscn 8339  ipfval 8348  sspval 8378  phop 8473  phpar 8479  h2hnm 8840  hhssnm 9126  hhsssh2 9135

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-nm 8215

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