Theorem phnv 30067

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 30066	. . 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 4229	. . 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 4017	. 2 ⊢ CPreHilOLD ⊆ NrmCVec
4	3	sseli 3979	1 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∩ cin 3948  ran crn 5678  ‘cfv 6544  (class class class)co 7409  {coprab 7410  1c1 11111   + caddc 11113   · cmul 11115  -cneg 11445  2c2 12267  ↑cexp 14027  NrmCVeccnv 29837  CPreHil_OLDccphlo 30065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-ph 30066

This theorem is referenced by:  phrel  30068  phnvi  30069  phop  30071  isph  30075  dipdi  30096  dipassr  30099  dipsubdir  30101  dipsubdi  30102  ajval  30114  minvecolem1  30127  minvecolem2  30128  minvecolem3  30129  minvecolem4a  30130  minvecolem4b  30131  minvecolem4  30133  minvecolem5  30134  minvecolem6  30135  minvecolem7  30136