MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phllmhm Structured version   Unicode version

Theorem phllmhm 18965
Description: The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
phllmhm.g  |-  G  =  ( x  e.  V  |->  ( x  .,  A
) )
Assertion
Ref Expression
phllmhm  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  G  e.  ( W LMHom  (ringLMod `  F
) ) )
Distinct variable groups:    x, A    x, 
.,    x, V    x, W
Allowed substitution hints:    F( x)    G( x)

Proof of Theorem phllmhm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . 5  |-  V  =  ( Base `  W
)
2 phlsrng.f . . . . 5  |-  F  =  (Scalar `  W )
3 phllmhm.h . . . . 5  |-  .,  =  ( .i `  W )
4 eqid 2402 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2402 . . . . 5  |-  ( *r `  F )  =  ( *r `  F )
6 eqid 2402 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18961 . . . 4  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. y  e.  V  (
( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
y  .,  y )  =  ( 0g `  F )  ->  y  =  ( 0g `  W ) )  /\  A. x  e.  V  ( ( *r `  F ) `  (
y  .,  x )
)  =  ( x 
.,  y ) ) ) )
87simp3bi 1014 . . 3  |-  ( W  e.  PreHil  ->  A. y  e.  V  ( ( x  e.  V  |->  ( x  .,  y ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( y  .,  y
)  =  ( 0g
`  F )  -> 
y  =  ( 0g
`  W ) )  /\  A. x  e.  V  ( ( *r `  F ) `
 ( y  .,  x ) )  =  ( x  .,  y
) ) )
9 simp1 997 . . . 4  |-  ( ( ( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
y  .,  y )  =  ( 0g `  F )  ->  y  =  ( 0g `  W ) )  /\  A. x  e.  V  ( ( *r `  F ) `  (
y  .,  x )
)  =  ( x 
.,  y ) )  ->  ( x  e.  V  |->  ( x  .,  y ) )  e.  ( W LMHom  (ringLMod `  F
) ) )
109ralimi 2797 . . 3  |-  ( A. y  e.  V  (
( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
y  .,  y )  =  ( 0g `  F )  ->  y  =  ( 0g `  W ) )  /\  A. x  e.  V  ( ( *r `  F ) `  (
y  .,  x )
)  =  ( x 
.,  y ) )  ->  A. y  e.  V  ( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
) )
118, 10syl 17 . 2  |-  ( W  e.  PreHil  ->  A. y  e.  V  ( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
) )
12 oveq2 6286 . . . . . 6  |-  ( y  =  A  ->  (
x  .,  y )  =  ( x  .,  A ) )
1312mpteq2dv 4482 . . . . 5  |-  ( y  =  A  ->  (
x  e.  V  |->  ( x  .,  y ) )  =  ( x  e.  V  |->  ( x 
.,  A ) ) )
14 phllmhm.g . . . . 5  |-  G  =  ( x  e.  V  |->  ( x  .,  A
) )
1513, 14syl6eqr 2461 . . . 4  |-  ( y  =  A  ->  (
x  e.  V  |->  ( x  .,  y ) )  =  G )
1615eleq1d 2471 . . 3  |-  ( y  =  A  ->  (
( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  <->  G  e.  ( W LMHom  (ringLMod `  F )
) ) )
1716rspccva 3159 . 2  |-  ( ( A. y  e.  V  ( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  A  e.  V )  ->  G  e.  ( W LMHom  (ringLMod `  F
) ) )
1811, 17sylan 469 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  G  e.  ( W LMHom  (ringLMod `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278   Basecbs 14841   *rcstv 14911  Scalarcsca 14912   .icip 14914   0gc0g 15054   *Ringcsr 17813   LMHom clmhm 17985   LVecclvec 18068  ringLModcrglmod 18135   PreHilcphl 18957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-iota 5533  df-fv 5577  df-ov 6281  df-phl 18959
This theorem is referenced by:  ipcl  18966  ip0l  18969  ipdir  18972  ipass  18978
  Copyright terms: Public domain W3C validator