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Theorem phlsrng 16863
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 2437 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 phlsrng.f . . 3  |-  F  =  (Scalar `  W )
3 eqid 2437 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2437 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2437 . . 3  |-  ( * r `  F )  =  ( * r `
 F )
6 eqid 2437 . . 3  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 16860 . 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 974 1  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   Basecbs 13470   * rcstv 13532  Scalarcsca 13533   .icip 13535   0gc0g 13724   *Ringcsr 15933   LMHom clmhm 16096   LVecclvec 16175  ringLModcrglmod 16242   PreHilcphl 16856
This theorem is referenced by:  iporthcom  16867  ip0r  16869  ipdi  16872  ip2di  16873  ipassr  16878  ipassr2  16879  cphcjcl  19147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-iota 5419  df-fv 5463  df-ov 6085  df-phl 16858
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