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Theorem cphsca 22157
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 2451 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2451 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2451 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 cphsca.f . . . 4  |-  F  =  (Scalar `  W )
5 cphsca.k . . . 4  |-  K  =  ( Base `  F
)
61, 2, 3, 4, 5iscph 22148 . . 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 1023 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
87simp3d 1022 1  |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 985    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404    |-> cmpt 4461   "cima 4837   ` cfv 5582  (class class class)co 6290   0cc0 9539   +oocpnf 9672   [,)cico 11637   sqrcsqrt 13296   Basecbs 15121   ↾s cress 15122  Scalarcsca 15193   .icip 15195  ℂfldccnfld 18970   PreHilcphl 19191   normcnm 21591  NrmModcnlm 21595   CPreHilccph 22144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fv 5590  df-ov 6293  df-cph 22146
This theorem is referenced by:  cphsubrg  22158  cphreccl  22159  cphcjcl  22161  cphqss  22166  cphclm  22167  ipcau  22212  hlprlem  22334  ishl2  22337
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