MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cphsca Structured version   Unicode version

Theorem cphsca 20823
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 20814 . . 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 1003 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
87simp3d 1002 1  |-  ( W  e.  CPreHil  ->  F  =  (flds  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3428    C_ wss 3429    |-> cmpt 4451   "cima 4944   ` cfv 5519  (class class class)co 6193   0cc0 9386   +oocpnf 9519   [,)cico 11406   sqrcsqr 12833   Basecbs 14285   ↾s cress 14286  Scalarcsca 14352   .icip 14354  ℂfldccnfld 17936   PreHilcphl 18171   normcnm 20294  NrmModcnlm 20298   CPreHilccph 20810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-xp 4947  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fv 5527  df-ov 6196  df-cph 20812
This theorem is referenced by:  cphsubrg  20824  cphreccl  20825  cphcjcl  20827  cphqss  20832  cphclm  20833  ipcau  20878  hlprlem  21004  ishl2  21007
  Copyright terms: Public domain W3C validator