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Theorem phllvec 16852
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 2435 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2435 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2435 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2435 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2435 . . 3  |-  ( * r `  (Scalar `  W ) )  =  ( * r `  (Scalar `  W ) )
6 eqid 2435 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 16851 . 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 1652    e. wcel 1725   A.wral 2697    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   Basecbs 13461   * rcstv 13523  Scalarcsca 13524   .icip 13526   0gc0g 13715   *Ringcsr 15924   LMHom clmhm 16087   LVecclvec 16166  ringLModcrglmod 16233   PreHilcphl 16847
This theorem is referenced by:  phllmod  16853  obsne0  16944  obslbs  16949  cphlvec  19130  tchclm  19181  ipcau2  19183  tchcph  19186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-iota 5410  df-fv 5454  df-ov 6076  df-phl 16849
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