Theorem phnv 30793

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 30792	. . 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 4196	. . 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 3990	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
4	3	sseli 3939	1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3910  ran crn 5632  ‘cfv 6499  (class class class)co 7369  {coprab 7370  1c1 11045   + caddc 11047   · cmul 11049  -cneg 11382  2c2 12217  ↑cexp 14002  NrmCVeccnv 30563  CPreHil_OLDccphlo 30791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-in 3918  df-ss 3928  df-ph 30792

This theorem is referenced by:  phrel  30794  phnvi  30795  phop  30797  isph  30801  dipdi  30822  dipassr  30825  dipsubdir  30827  dipsubdi  30828  ajval  30840  minvecolem1  30853  minvecolem2  30854  minvecolem3  30855  minvecolem4a  30856  minvecolem4b  30857  minvecolem4  30859  minvecolem5  30860  minvecolem6  30861  minvecolem7  30862