Theorem phnvi 31016

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 31014	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

Syntax hints:   ∈ wcel 2142  NrmCVeccnv 30784  CPreHil_OLDccphlo 31012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-ss 3921  df-ph 31013

This theorem is referenced by:  elimph  31020  ip0i  31025  ip1ilem  31026  ip2i  31028  ipdirilem  31029  ipasslem1  31031  ipasslem2  31032  ipasslem4  31034  ipasslem5  31035  ipasslem7  31036  ipasslem8  31037  ipasslem9  31038  ipasslem10  31039  ipasslem11  31040  ip2dii  31044  pythi  31050  siilem1  31051  siilem2  31052  siii  31053  ipblnfi  31055  ip2eqi  31056  ajfuni  31059