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Theorem phllvec 16533
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 2283 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2283 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2283 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2283 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2283 . . 3  |-  ( * r `  (Scalar `  W ) )  =  ( * r `  (Scalar `  W ) )
6 eqid 2283 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 16532 . 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 970 1  |-  ( W  e.  PreHil  ->  W  e.  LVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Basecbs 13148   * rcstv 13210  Scalarcsca 13211   .icip 13213   0gc0g 13400   *Ringcsr 15609   LMHom clmhm 15776   LVecclvec 15855  ringLModcrglmod 15922   PreHilcphl 16528
This theorem is referenced by:  phllmod  16534  obsne0  16625  obslbs  16630  cphlvec  18611  tchclm  18662  ipcau2  18664  tchcph  18667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-iota 5219  df-fv 5263  df-ov 5861  df-phl 16530
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