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

Theorem phlsrng 18856
Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypothesis
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
phlsrng  |-  ( W  e.  PreHil  ->  F  e.  *Ring )

Proof of Theorem phlsrng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 phlsrng.f . . 3  |-  F  =  (Scalar `  W )
3 eqid 2402 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2402 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2402 . . 3  |-  ( *r `  F )  =  ( *r `  F )
6 eqid 2402 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18853 . 2  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  ( Base `  W ) ( ( y  e.  ( Base `  W )  |->  ( y ( .i `  W
) x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x ( .i `  W ) x )  =  ( 0g `  F )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  ( Base `  W
) ( ( *r `  F ) `
 ( x ( .i `  W ) y ) )  =  ( y ( .i
`  W ) x ) ) ) )
87simp2bi 1013 1  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753    |-> cmpt 4452   ` cfv 5525  (class class class)co 6234   Basecbs 14733   *rcstv 14803  Scalarcsca 14804   .icip 14806   0gc0g 14946   *Ringcsr 17705   LMHom clmhm 17877   LVecclvec 17960  ringLModcrglmod 18027   PreHilcphl 18849
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-iota 5489  df-fv 5533  df-ov 6237  df-phl 18851
This theorem is referenced by:  iporthcom  18860  ip0r  18862  ipdi  18865  ip2di  18866  ipassr  18871  ipassr2  18872  cphcjcl  21814
  Copyright terms: Public domain W3C validator