Theorem phnv 30889

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 30888	. . 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 3980	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
4	3	sseli 3929	1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900  ran crn 5625  ‘cfv 6492  (class class class)co 7358  {coprab 7359  1c1 11027   + caddc 11029   · cmul 11031  -cneg 11365  2c2 12200  ↑cexp 13984  NrmCVeccnv 30659  CPreHil_OLDccphlo 30887

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 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-in 3908  df-ss 3918  df-ph 30888

This theorem is referenced by:  phrel  30890  phnvi  30891  phop  30893  isph  30897  dipdi  30918  dipassr  30921  dipsubdir  30923  dipsubdi  30924  ajval  30936  minvecolem1  30949  minvecolem2  30950  minvecolem3  30951  minvecolem4a  30952  minvecolem4b  30953  minvecolem4  30955  minvecolem5  30956  minvecolem6  30957  minvecolem7  30958