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Theorem phllvec 18193
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 2454 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2454 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2454 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2454 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2454 . . 3  |-  ( *r `  (Scalar `  W ) )  =  ( *r `  (Scalar `  W ) )
6 eqid 2454 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 18192 . 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 1003 1  |-  ( W  e.  PreHil  ->  W  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   Basecbs 14296   *rcstv 14363  Scalarcsca 14364   .icip 14366   0gc0g 14501   *Ringcsr 17062   LMHom clmhm 17233   LVecclvec 17316  ringLModcrglmod 17383   PreHilcphl 18188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4532
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-iota 5492  df-fv 5537  df-ov 6206  df-phl 18190
This theorem is referenced by:  phllmod  18194  obsne0  18285  obslbs  18290  cphlvec  20836  tchclm  20889  ipcau2  20891  tchcph  20894
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