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Theorem phllvec 18033
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 2438 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2438 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2438 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2438 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2438 . . 3  |-  ( *r `  (Scalar `  W ) )  =  ( *r `  (Scalar `  W ) )
6 eqid 2438 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 18032 . 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 1369    e. wcel 1756   A.wral 2710    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   Basecbs 14166   *rcstv 14232  Scalarcsca 14233   .icip 14235   0gc0g 14370   *Ringcsr 16907   LMHom clmhm 17077   LVecclvec 17160  ringLModcrglmod 17227   PreHilcphl 18028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-iota 5376  df-fv 5421  df-ov 6089  df-phl 18030
This theorem is referenced by:  phllmod  18034  obsne0  18125  obslbs  18130  cphlvec  20669  tchclm  20722  ipcau2  20724  tchcph  20727
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