Theorem phnv 8469

Description: Every complex inner product space is a normed complex vector space.

Assertion

Ref	Expression
phnv	|- (U e. CPreHil -> U e. NrmCVec)

Proof of Theorem phnv

Step	Hyp	Ref	Expression
1		df-ph 8468	. . 3 |- CPreHil = (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))})
2		inss1 2233	. . 3 |- (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}) (_ NrmCVec
3	1, 2	eqsstr 2094	. 2 |- CPreHil (_ NrmCVec
4	3	sseli 2068	1 |- (U e. CPreHil -> U e. NrmCVec)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  A.wral 1648   i^i cin 2049  ran crn 3177  ` cfv 3188  (class class class)co 3969  {copab2 3970  1c1 5247   + caddc 5249   x. cmul 5251  -ucneg 5305  2c2 5963  ^cexp 6569  NrmCVeccnv 8199  CPreHilcphl 8467

This theorem is referenced by:  phrel 8470  phnvi 8471  phop 8473  isph 8477  ipdi 8499  ipassr 8502  ipsubdir 8504  ipsubdi 8505  sspph 8511

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-in 2054  df-ss 2056  df-ph 8468

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