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Theorem cphsca 21916
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 2402 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2402 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2402 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 cphsca.f . . . 4  |-  F  =  (Scalar `  W )
5 cphsca.k . . . 4  |-  K  =  ( Base `  F
)
61, 2, 3, 4, 5iscph 21907 . . 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 1012 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
87simp3d 1011 1  |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413    |-> cmpt 4452   "cima 4825   ` cfv 5568  (class class class)co 6277   0cc0 9521   +oocpnf 9654   [,)cico 11583   sqrcsqrt 13213   Basecbs 14839   ↾s cress 14840  Scalarcsca 14910   .icip 14912  ℂfldccnfld 18738   PreHilcphl 18955   normcnm 21387  NrmModcnlm 21391   CPreHilccph 21903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-xp 4828  df-cnv 4830  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fv 5576  df-ov 6280  df-cph 21905
This theorem is referenced by:  cphsubrg  21917  cphreccl  21918  cphcjcl  21920  cphqss  21925  cphclm  21926  ipcau  21971  hlprlem  22097  ishl2  22100
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