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Theorem phllvec 18533
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 2467 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2467 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2467 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2467 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2467 . . 3  |-  ( *r `  (Scalar `  W ) )  =  ( *r `  (Scalar `  W ) )
6 eqid 2467 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 18532 . 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 1011 1  |-  ( W  e.  PreHil  ->  W  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   Basecbs 14507   *rcstv 14574  Scalarcsca 14575   .icip 14577   0gc0g 14712   *Ringcsr 17364   LMHom clmhm 17536   LVecclvec 17619  ringLModcrglmod 17686   PreHilcphl 18528
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 4582
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-iota 5557  df-fv 5602  df-ov 6298  df-phl 18530
This theorem is referenced by:  phllmod  18534  obsne0  18625  obslbs  18630  cphlvec  21490  tchclm  21543  ipcau2  21545  tchcph  21548
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