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Theorem cphsca 19132
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 2435 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2435 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2435 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 cphsca.f . . . 4  |-  F  =  (Scalar `  W )
5 cphsca.k . . . 4  |-  K  =  ( Base `  F
)
61, 2, 3, 4, 5iscph 19123 . . 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 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312    e. cmpt 4258   "cima 4873   ` cfv 5446  (class class class)co 6073   0cc0 8980    +oocpnf 9107   [,)cico 10908   sqrcsqr 12028   Basecbs 13459   ↾s cress 13460  Scalarcsca 13522   .icip 13524  ℂfldccnfld 16693   PreHilcphl 16845   normcnm 18614  NrmModcnlm 18618   CPreHilccph 19119
This theorem is referenced by:  cphsubrg  19133  cphreccl  19134  cphcjcl  19136  cphqss  19141  cphclm  19142  ipcau  19185  hlprlem  19311  ishl2  19314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-cph 19121
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