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Theorem cphsca 19006
Description: A complex pre-Hilbert space is a vector space over a subfield of  CC. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
cphsca.f  |-  F  =  (Scalar `  W )
cphsca.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
cphsca  |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )

Proof of Theorem cphsca
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2380 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2380 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2380 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 cphsca.f . . . 4  |-  F  =  (Scalar `  W )
5 cphsca.k . . . 4  |-  K  =  ( Base `  F
)
61, 2, 3, 4, 5iscph 18997 . . 3  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( sqr " ( K  i^i  ( 0 [,)  +oo ) ) )  C_  K  /\  ( norm `  W
)  =  ( x  e.  ( Base `  W
)  |->  ( sqr `  (
x ( .i `  W ) x ) ) ) ) )
76simp1bi 972 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
87simp3d 971 1  |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3255    C_ wss 3256    e. cmpt 4200   "cima 4814   ` cfv 5387  (class class class)co 6013   0cc0 8916    +oocpnf 9043   [,)cico 10843   sqrcsqr 11958   Basecbs 13389   ↾s cress 13390  Scalarcsca 13452   .icip 13454  ℂfldccnfld 16619   PreHilcphl 16771   normcnm 18488  NrmModcnlm 18492   CPreHilccph 18993
This theorem is referenced by:  cphsubrg  19007  cphreccl  19008  cphcjcl  19010  cphqss  19015  cphclm  19016  ipcau  19059  hlprlem  19181  ishl2  19184
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-nul 4272
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 2235  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fv 5395  df-ov 6016  df-cph 18995
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