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Theorem cphphl 19139
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 2438 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2438 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2438 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2438 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2438 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 19138 . . 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 973 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp1d 970 1  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322    e. cmpt 4269   "cima 4884   ` cfv 5457  (class class class)co 6084   0cc0 8995    +oocpnf 9122   [,)cico 10923   sqrcsqr 12043   Basecbs 13474   ↾s cress 13475  Scalarcsca 13537   .icip 13539  ℂfldccnfld 16708   PreHilcphl 16860   normcnm 18629  NrmModcnlm 18633   CPreHilccph 19134
This theorem is referenced by:  cphlvec  19143  cphcjcl  19151  cphipcl  19159  cphnmf  19163  cphipcj  19167  cphorthcom  19168  cphip0l  19169  cphip0r  19170  cphipeq0  19171  cphdir  19172  cphdi  19173  cph2di  19174  cphsubdir  19175  cphsubdi  19176  cph2subdi  19177  cphass  19178  cphassr  19179  ipcau  19200  nmparlem  19201  ipcn  19205  hlphl  19324  pjthlem2  19344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-cph 19136
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