Theorem phnvi 30752

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 30750	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2109  NrmCVeccnv 30520  CPreHil_OLDccphlo 30748

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 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-ss 3934  df-ph 30749

This theorem is referenced by:  elimph  30756  ip0i  30761  ip1ilem  30762  ip2i  30764  ipdirilem  30765  ipasslem1  30767  ipasslem2  30768  ipasslem4  30770  ipasslem5  30771  ipasslem7  30772  ipasslem8  30773  ipasslem9  30774  ipasslem10  30775  ipasslem11  30776  ip2dii  30780  pythi  30786  siilem1  30787  siilem2  30788  siii  30789  ipblnfi  30791  ip2eqi  30792  ajfuni  30795