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Theorem phlsrng 19198
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 19195 . 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 1024 1  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   Basecbs 15121   *rcstv 15192  Scalarcsca 15193   .icip 15195   0gc0g 15338   *Ringcsr 18072   LMHom clmhm 18242   LVecclvec 18325  ringLModcrglmod 18392   PreHilcphl 19191
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-iota 5546  df-fv 5590  df-ov 6293  df-phl 19193
This theorem is referenced by:  iporthcom  19202  ip0r  19204  ipdi  19207  ip2di  19208  ipassr  19213  ipassr2  19214  cphcjcl  22161
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