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Theorem phlsrng 18171
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 2451 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 phlsrng.f . . 3  |-  F  =  (Scalar `  W )
3 eqid 2451 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2451 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2451 . . 3  |-  ( *r `  F )  =  ( *r `  F )
6 eqid 2451 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18168 . 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 1004 1  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    |-> cmpt 4450   ` cfv 5518  (class class class)co 6192   Basecbs 14278   *rcstv 14344  Scalarcsca 14345   .icip 14347   0gc0g 14482   *Ringcsr 17037   LMHom clmhm 17208   LVecclvec 17291  ringLModcrglmod 17358   PreHilcphl 18164
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 4521
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 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-iota 5481  df-fv 5526  df-ov 6195  df-phl 18166
This theorem is referenced by:  iporthcom  18175  ip0r  18177  ipdi  18180  ip2di  18181  ipassr  18186  ipassr2  18187  cphcjcl  20820
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