Theorem phnvi 28596

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 28594	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2113  NrmCVeccnv 28364  CPreHil_OLDccphlo 28592

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-v 3499  df-in 3946  df-ss 3955  df-ph 28593

This theorem is referenced by:  elimph  28600  ip0i  28605  ip1ilem  28606  ip2i  28608  ipdirilem  28609  ipasslem1  28611  ipasslem2  28612  ipasslem4  28614  ipasslem5  28615  ipasslem7  28616  ipasslem8  28617  ipasslem9  28618  ipasslem10  28619  ipasslem11  28620  ip2dii  28624  pythi  28630  siilem1  28631  siilem2  28632  siii  28633  ipblnfi  28635  ip2eqi  28636  ajfuni  28639