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Theorem cphphl 20665
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 20664 . . 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 1369    e. wcel 1756    i^i cin 3322    C_ wss 3323    e. cmpt 4345   "cima 4838   ` cfv 5413  (class class class)co 6086   0cc0 9274   +oocpnf 9407   [,)cico 11294   sqrcsqr 12714   Basecbs 14166   ↾s cress 14167  Scalarcsca 14233   .icip 14235  ℂfldccnfld 17793   PreHilcphl 18028   normcnm 20144  NrmModcnlm 20148   CPreHilccph 20660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-xp 4841  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-ov 6089  df-cph 20662
This theorem is referenced by:  cphlvec  20669  cphcjcl  20677  cphipcl  20685  cphnmf  20689  cphipcj  20693  cphorthcom  20694  cphip0l  20695  cphip0r  20696  cphipeq0  20697  cphdir  20698  cphdi  20699  cph2di  20700  cphsubdir  20701  cphsubdi  20702  cph2subdi  20703  cphass  20704  cphassr  20705  ipcau  20728  nmparlem  20729  ipcn  20733  hlphl  20852  pjthlem2  20900
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