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Theorem phllvec 16549
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 2296 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2296 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
4 eqid 2296 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2296 . . 3  |-  ( * r `  (Scalar `  W ) )  =  ( * r `  (Scalar `  W ) )
6 eqid 2296 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
71, 2, 3, 4, 5, 6isphl 16548 . 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 1632    e. wcel 1696   A.wral 2556    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Basecbs 13164   * rcstv 13226  Scalarcsca 13227   .icip 13229   0gc0g 13416   *Ringcsr 15625   LMHom clmhm 15792   LVecclvec 15871  ringLModcrglmod 15938   PreHilcphl 16544
This theorem is referenced by:  phllmod  16550  obsne0  16641  obslbs  16646  cphlvec  18627  tchclm  18678  ipcau2  18680  tchcph  18683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-iota 5235  df-fv 5279  df-ov 5877  df-phl 16546
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