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Theorem phlsrng 18433
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 2467 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 phlsrng.f . . 3  |-  F  =  (Scalar `  W )
3 eqid 2467 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2467 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2467 . . 3  |-  ( *r `  F )  =  ( *r `  F )
6 eqid 2467 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18430 . 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 1012 1  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   Basecbs 14486   *rcstv 14553  Scalarcsca 14554   .icip 14556   0gc0g 14691   *Ringcsr 17276   LMHom clmhm 17448   LVecclvec 17531  ringLModcrglmod 17598   PreHilcphl 18426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-iota 5549  df-fv 5594  df-ov 6285  df-phl 18428
This theorem is referenced by:  iporthcom  18437  ip0r  18439  ipdi  18442  ip2di  18443  ipassr  18448  ipassr2  18449  cphcjcl  21365
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