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

Theorem cphphl 21350
Description: A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphphl  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )

Proof of Theorem cphphl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2467 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2467 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2467 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 21349 . . 3  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) )  /\  ( sqr " ( (
Base `  (Scalar `  W
) )  i^i  (
0 [,) +oo )
) )  C_  ( Base `  (Scalar `  W
) )  /\  ( norm `  W )  =  ( x  e.  (
Base `  W )  |->  ( sqr `  (
x ( .i `  W ) x ) ) ) ) )
76simp1bi 1011 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp1d 1008 1  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476    |-> cmpt 4505   "cima 5002   ` cfv 5586  (class class class)co 6282   0cc0 9488   +oocpnf 9621   [,)cico 11527   sqrcsqrt 13023   Basecbs 14483   ↾s cress 14484  Scalarcsca 14551   .icip 14553  ℂfldccnfld 18188   PreHilcphl 18423   normcnm 20829  NrmModcnlm 20833   CPreHilccph 21345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-ov 6285  df-cph 21347
This theorem is referenced by:  cphlvec  21354  cphcjcl  21362  cphipcl  21370  cphnmf  21374  cphipcj  21378  cphorthcom  21379  cphip0l  21380  cphip0r  21381  cphipeq0  21382  cphdir  21383  cphdi  21384  cph2di  21385  cphsubdir  21386  cphsubdi  21387  cph2subdi  21388  cphass  21389  cphassr  21390  ipcau  21413  nmparlem  21414  ipcn  21418  hlphl  21537  pjthlem2  21585
  Copyright terms: Public domain W3C validator