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Theorem cphphl 20821
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 2454 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2454 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2454 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2454 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2454 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 20820 . . 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 1003 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp1d 1000 1  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3434    C_ wss 3435    |-> cmpt 4457   "cima 4950   ` cfv 5525  (class class class)co 6199   0cc0 9392   +oocpnf 9525   [,)cico 11412   sqrcsqr 12839   Basecbs 14291   ↾s cress 14292  Scalarcsca 14359   .icip 14361  ℂfldccnfld 17942   PreHilcphl 18177   normcnm 20300  NrmModcnlm 20304   CPreHilccph 20816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-xp 4953  df-cnv 4955  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fv 5533  df-ov 6202  df-cph 20818
This theorem is referenced by:  cphlvec  20825  cphcjcl  20833  cphipcl  20841  cphnmf  20845  cphipcj  20849  cphorthcom  20850  cphip0l  20851  cphip0r  20852  cphipeq0  20853  cphdir  20854  cphdi  20855  cph2di  20856  cphsubdir  20857  cphsubdi  20858  cph2subdi  20859  cphass  20860  cphassr  20861  ipcau  20884  nmparlem  20885  ipcn  20889  hlphl  21008  pjthlem2  21056
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