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Theorem cphphl 19087
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 2404 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2404 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2404 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2404 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2404 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 19086 . . 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 972 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp1d 969 1  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280    e. cmpt 4226   "cima 4840   ` cfv 5413  (class class class)co 6040   0cc0 8946    +oocpnf 9073   [,)cico 10874   sqrcsqr 11993   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .icip 13489  ℂfldccnfld 16658   PreHilcphl 16810   normcnm 18577  NrmModcnlm 18581   CPreHilccph 19082
This theorem is referenced by:  cphlvec  19091  cphcjcl  19099  cphipcl  19107  cphnmf  19111  cphipcj  19115  cphorthcom  19116  cphip0l  19117  cphip0r  19118  cphipeq0  19119  cphdir  19120  cphdi  19121  cph2di  19122  cphsubdir  19123  cphsubdi  19124  cph2subdi  19125  cphass  19126  cphassr  19127  ipcau  19148  nmparlem  19149  ipcn  19153  hlphl  19272  pjthlem2  19292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-cph 19084
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