MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phrel Structured version   Unicode version

Theorem phrel 26441
Description: The class of all complex inner product spaces is a relation. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
phrel  |-  Rel  CPreHil OLD

Proof of Theorem phrel
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 phnv 26440 . . 3  |-  ( x  e.  CPreHil OLD  ->  x  e.  NrmCVec )
21ssriv 3468 . 2  |-  CPreHil OLD  C_  NrmCVec
3 nvrel 26206 . 2  |-  Rel  NrmCVec
4 relss 4937 . 2  |-  ( CPreHil OLD  C_  NrmCVec  ->  ( Rel  NrmCVec  ->  Rel  CPreHil OLD ) )
52, 3, 4mp2 9 1  |-  Rel  CPreHil OLD
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3436   Rel wrel 4854   NrmCVeccnv 26188   CPreHil OLDccphlo 26438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-opab 4480  df-xp 4855  df-rel 4856  df-oprab 6305  df-nv 26196  df-ph 26439
This theorem is referenced by:  phop  26444
  Copyright terms: Public domain W3C validator