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Theorem phllvec 16815
Description: A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
phllvec  |-  ( W  e.  PreHil  ->  W  e.  LVec )

Proof of Theorem phllvec
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2404 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2404 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2404 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2404 . . 3  |-  ( * r `  (Scalar `  W ) )  =  ( * r `  (Scalar `  W ) )
6 eqid 2404 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 16814 . 2  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  (Scalar `  W )  e.  *Ring  /\  A. x  e.  ( Base `  W
) ( ( y  e.  ( Base `  W
)  |->  ( y ( .i `  W ) x ) )  e.  ( W LMHom  (ringLMod `  (Scalar `  W ) ) )  /\  ( ( x ( .i `  W
) x )  =  ( 0g `  (Scalar `  W ) )  ->  x  =  ( 0g `  W ) )  /\  A. y  e.  ( Base `  W ) ( ( * r `  (Scalar `  W ) ) `  ( x ( .i
`  W ) y ) )  =  ( y ( .i `  W ) x ) ) ) )
87simp1bi 972 1  |-  ( W  e.  PreHil  ->  W  e.  LVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   Basecbs 13424   * rcstv 13486  Scalarcsca 13487   .icip 13489   0gc0g 13678   *Ringcsr 15887   LMHom clmhm 16050   LVecclvec 16129  ringLModcrglmod 16196   PreHilcphl 16810
This theorem is referenced by:  phllmod  16816  obsne0  16907  obslbs  16912  cphlvec  19091  tchclm  19142  ipcau2  19144  tchcph  19147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-iota 5377  df-fv 5421  df-ov 6043  df-phl 16812
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