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Theorem phllvec 18531
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 2441 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2441 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2441 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2441 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2441 . . 3  |-  ( *r `  (Scalar `  W ) )  =  ( *r `  (Scalar `  W ) )
6 eqid 2441 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 18530 . 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 1010 1  |-  ( W  e.  PreHil  ->  W  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   Basecbs 14504   *rcstv 14571  Scalarcsca 14572   .icip 14574   0gc0g 14709   *Ringcsr 17361   LMHom clmhm 17533   LVecclvec 17616  ringLModcrglmod 17683   PreHilcphl 18526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-iota 5537  df-fv 5582  df-ov 6280  df-phl 18528
This theorem is referenced by:  phllmod  18532  obsne0  18623  obslbs  18628  cphlvec  21488  tchclm  21541  ipcau2  21543  tchcph  21546
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