Theorem phnvi 31475

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 31473	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
3	1, 2	ax-mp 5	1 ⊢ 𝑈 ∈ NrmCVec

This proof depends on syntax axioms:   ∈ wcel 2147  NrmCVeccnv 31243  CPreHil_OLDccphlo 31471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-ph 31472

This theorem is used by:  elimph  31479  ip0i  31484  ip1ilem  31485  ip2i  31487  ipdirilem  31488  ipasslem1  31490  ipasslem2  31491  ipasslem4  31493  ipasslem5  31494  ipasslem7  31495  ipasslem8  31496  ipasslem9  31497  ipasslem10  31498  ipasslem11  31499  ip2dii  31503  pythi  31509  siilem1  31510  siilem2  31511  siii  31512  ipblnfi  31514  ip2eqi  31515  ajfuni  31518