Theorem phnvi 27671

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 e. CPreHil OLD

Assertion

Ref	Expression
phnvi	|- U e. NrmCVec

Proof of Theorem phnvi

Step	Hyp	Ref	Expression
1		phnvi.1	. 2 |- U e. CPreHil OLD
2		phnv 27669	. 2 |- ( U e. CPreHil OLD -> U e. NrmCVec )
3	1, 2	ax-mp 5	1 |- U e. NrmCVec

Syntax hints:   e. wcel 1990   NrmCVeccnv 27439   CPreHil OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-ph 27668

This theorem is referenced by:  elimph  27675  ip0i  27680  ip1ilem  27681  ip2i  27683  ipdirilem  27684  ipasslem1  27686  ipasslem2  27687  ipasslem4  27689  ipasslem5  27690  ipasslem7  27691  ipasslem8  27692  ipasslem9  27693  ipasslem10  27694  ipasslem11  27695  ip2dii  27699  pythi  27705  siilem1  27706  siilem2  27707  siii  27708  ipblnfi  27711  ip2eqi  27712  ajfuni  27715