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Theorem phllmhm 18059
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 2441 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
5 eqid 2441 . . . . 5  |-  ( *r `  F )  =  ( *r `  F )
6 eqid 2441 . . . . 5  |-  ( 0g
`  F )  =  ( 0g `  F
)
71, 2, 3, 4, 5, 6isphl 18055 . . . 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 1005 . . 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 988 . . . 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 2789 . . 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 16 . 2  |-  ( W  e.  PreHil  ->  A. y  e.  V  ( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
) )
12 oveq2 6097 . . . . . 6  |-  ( y  =  A  ->  (
x  .,  y )  =  ( x  .,  A ) )
1312mpteq2dv 4377 . . . . 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 2491 . . . 4  |-  ( y  =  A  ->  (
x  e.  V  |->  ( x  .,  y ) )  =  G )
1615eleq1d 2507 . . 3  |-  ( y  =  A  ->  (
( x  e.  V  |->  ( x  .,  y
) )  e.  ( W LMHom  (ringLMod `  F )
)  <->  G  e.  ( W LMHom  (ringLMod `  F )
) ) )
1716rspccva 3070 . 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 471 1  |-  ( ( W  e.  PreHil  /\  A  e.  V )  ->  G  e.  ( W LMHom  (ringLMod `  F
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713    e. cmpt 4348   ` cfv 5416  (class class class)co 6089   Basecbs 14172   *rcstv 14238  Scalarcsca 14239   .icip 14241   0gc0g 14376   *Ringcsr 16927   LMHom clmhm 17098   LVecclvec 17181  ringLModcrglmod 17248   PreHilcphl 18051
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 2422  ax-nul 4419
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 2257  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-iota 5379  df-fv 5424  df-ov 6092  df-phl 18053
This theorem is referenced by:  ipcl  18060  ip0l  18063  ipdir  18066  ipass  18072
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