Theorem phnv 31166

Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
phnv	⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Proof of Theorem phnv

Dummy variables 𝑔 𝑛 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ph 31165	. . 3 ⊢ CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))})
2		inss1 4189	. . 3 ⊢ (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) ⊆ NrmCVec
3	1, 2	eqsstri 3983	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
4	3	sseli 3933	1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904  ran crn 5662  ‘cfv 6536  (class class class)co 7410  {coprab 7411  1c1 11096   + caddc 11098   · cmul 11100  -cneg 11437  2c2 12290  ↑cexp 14093  NrmCVeccnv 30936  CPreHil_OLDccphlo 31164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3912  df-ss 3922  df-ph 31165

This theorem is referenced by:  phrel  31167  phnvi  31168  phop  31170  isph  31174  dipdi  31195  dipassr  31198  dipsubdir  31200  dipsubdi  31201  ajval  31213  minvecolem1  31226  minvecolem2  31227  minvecolem3  31228  minvecolem4a  31229  minvecolem4b  31230  minvecolem4  31232  minvecolem5  31233  minvecolem6  31234  minvecolem7  31235