Theorem phnv 30794

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 30793	. . 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 4184	. . 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 3976	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
4	3	sseli 3925	1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∩ cin 3896  ran crn 5615  ‘cfv 6481  (class class class)co 7346  {coprab 7347  1c1 11007   + caddc 11009   · cmul 11011  -cneg 11345  2c2 12180  ↑cexp 13968  NrmCVeccnv 30564  CPreHil_OLDccphlo 30792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-ss 3914  df-ph 30793

This theorem is referenced by:  phrel  30795  phnvi  30796  phop  30798  isph  30802  dipdi  30823  dipassr  30826  dipsubdir  30828  dipsubdi  30829  ajval  30841  minvecolem1  30854  minvecolem2  30855  minvecolem3  30856  minvecolem4a  30857  minvecolem4b  30858  minvecolem4  30860  minvecolem5  30861  minvecolem6  30862  minvecolem7  30863