Theorem phnvi 28196

Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
phnvi.1	⊢ 𝑈 ∈ CPreHilOLD

Assertion

Ref	Expression
phnvi	⊢ 𝑈 ∈ NrmCVec

Proof of Theorem phnvi

Step	Hyp	Ref	Expression
1		phnvi.1	. 2 ⊢ 𝑈 ∈ CPreHilOLD
2		phnv 28194	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2157  NrmCVeccnv 27964  CPreHil_OLDccphlo 28192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-ph 28193

This theorem is referenced by:  elimph  28200  ip0i  28205  ip1ilem  28206  ip2i  28208  ipdirilem  28209  ipasslem1  28211  ipasslem2  28212  ipasslem4  28214  ipasslem5  28215  ipasslem7  28216  ipasslem8  28217  ipasslem9  28218  ipasslem10  28219  ipasslem11  28220  ip2dii  28224  pythi  28230  siilem1  28231  siilem2  28232  siii  28233  ipblnfi  28236  ip2eqi  28237  ajfuni  28240