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Theorem isphl 16822
Description: The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
isphl.v  |-  V  =  ( Base `  W
)
isphl.f  |-  F  =  (Scalar `  W )
isphl.h  |-  .,  =  ( .i `  W )
isphl.o  |-  .0.  =  ( 0g `  W )
isphl.i  |-  .*  =  ( * r `  F )
isphl.z  |-  Z  =  ( 0g `  F
)
Assertion
Ref Expression
isphl  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y  .,  x
) ) ) )
Distinct variable groups:    x, y, V    x, W, y
Allowed substitution hints:    F( x, y)    ., ( x, y)    .* ( x, y)    .0. ( x, y)    Z( x, y)

Proof of Theorem isphl
Dummy variables  f 
g  h  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5709 . . . . 5  |-  ( Base `  g )  e.  _V
21a1i 11 . . . 4  |-  ( g  =  W  ->  ( Base `  g )  e. 
_V )
3 fvex 5709 . . . . . 6  |-  ( .i
`  g )  e. 
_V
43a1i 11 . . . . 5  |-  ( ( g  =  W  /\  v  =  ( Base `  g ) )  -> 
( .i `  g
)  e.  _V )
5 fvex 5709 . . . . . . 7  |-  (Scalar `  g )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  e.  _V )
7 id 20 . . . . . . . . 9  |-  ( f  =  (Scalar `  g
)  ->  f  =  (Scalar `  g ) )
8 simpll 731 . . . . . . . . . . 11  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  g  =  W )
98fveq2d 5699 . . . . . . . . . 10  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  =  (Scalar `  W ) )
10 isphl.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
119, 10syl6eqr 2462 . . . . . . . . 9  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  =  F )
127, 11sylan9eqr 2466 . . . . . . . 8  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  f  =  F )
1312eleq1d 2478 . . . . . . 7  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
f  e.  *Ring  <->  F  e.  *Ring
) )
14 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  v  =  ( Base `  g
) )
15 simplll 735 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  g  =  W )
1615fveq2d 5699 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( Base `  g )  =  ( Base `  W
) )
17 isphl.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
1816, 17syl6eqr 2462 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( Base `  g )  =  V )
1914, 18eqtrd 2444 . . . . . . . 8  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  v  =  V )
20 simplr 732 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  h  =  ( .i `  g ) )
2115fveq2d 5699 . . . . . . . . . . . . . 14  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( .i `  g )  =  ( .i `  W
) )
22 isphl.h . . . . . . . . . . . . . 14  |-  .,  =  ( .i `  W )
2321, 22syl6eqr 2462 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( .i `  g )  = 
.,  )
2420, 23eqtrd 2444 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  h  =  .,  )
2524oveqd 6065 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
y h x )  =  ( y  .,  x ) )
2619, 25mpteq12dv 4255 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
y  e.  v  |->  ( y h x ) )  =  ( y  e.  V  |->  ( y 
.,  x ) ) )
2712fveq2d 5699 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (ringLMod `  f )  =  (ringLMod `  F ) )
2815, 27oveq12d 6066 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
g LMHom  (ringLMod `  f )
)  =  ( W LMHom 
(ringLMod `  F ) ) )
2926, 28eleq12d 2480 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( y  e.  v 
|->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f )
)  <->  ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) ) ) )
3024oveqd 6065 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x h x )  =  ( x  .,  x ) )
3112fveq2d 5699 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  f )  =  ( 0g `  F
) )
32 isphl.z . . . . . . . . . . . 12  |-  Z  =  ( 0g `  F
)
3331, 32syl6eqr 2462 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  f )  =  Z )
3430, 33eqeq12d 2426 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( x h x )  =  ( 0g
`  f )  <->  ( x  .,  x )  =  Z ) )
3515fveq2d 5699 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  g )  =  ( 0g `  W
) )
36 isphl.o . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  W )
3735, 36syl6eqr 2462 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  g )  =  .0.  )
3837eqeq2d 2423 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x  =  ( 0g
`  g )  <->  x  =  .0.  ) )
3934, 38imbi12d 312 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( ( x h x )  =  ( 0g `  f )  ->  x  =  ( 0g `  g ) )  <->  ( ( x 
.,  x )  =  Z  ->  x  =  .0.  ) ) )
4012fveq2d 5699 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
* r `  f
)  =  ( * r `  F ) )
41 isphl.i . . . . . . . . . . . . 13  |-  .*  =  ( * r `  F )
4240, 41syl6eqr 2462 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
* r `  f
)  =  .*  )
4324oveqd 6065 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x h y )  =  ( x  .,  y ) )
4442, 43fveq12d 5701 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( * r `  f ) `  (
x h y ) )  =  (  .* 
`  ( x  .,  y ) ) )
4544, 25eqeq12d 2426 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( ( * r `
 f ) `  ( x h y ) )  =  ( y h x )  <-> 
(  .*  `  (
x  .,  y )
)  =  ( y 
.,  x ) ) )
4619, 45raleqbidv 2884 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( A. y  e.  v 
( ( * r `
 f ) `  ( x h y ) )  =  ( y h x )  <->  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) )
4729, 39, 463anbi123d 1254 . . . . . . . 8  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f
) )  /\  (
( x h x )  =  ( 0g
`  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) )  <-> 
( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) ) )
4819, 47raleqbidv 2884 . . . . . . 7  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( A. x  e.  v 
( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f
) )  /\  (
( x h x )  =  ( 0g
`  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) )  <->  A. x  e.  V  ( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) ) )
4913, 48anbi12d 692 . . . . . 6  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( f  e.  *Ring  /\ 
A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f
) )  /\  (
( x h x )  =  ( 0g
`  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) ) )  <->  ( F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
.,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y
) )  =  ( y  .,  x ) ) ) ) )
506, 49sbcied 3165 . . . . 5  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  ( [. (Scalar `  g )  / 
f ]. ( f  e.  *Ring  /\  A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f ) )  /\  ( ( x h x )  =  ( 0g `  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( (
* r `  f
) `  ( x h y ) )  =  ( y h x ) ) )  <-> 
( F  e.  *Ring  /\ 
A. x  e.  V  ( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) ) ) )
514, 50sbcied 3165 . . . 4  |-  ( ( g  =  W  /\  v  =  ( Base `  g ) )  -> 
( [. ( .i `  g )  /  h ]. [. (Scalar `  g
)  /  f ]. ( f  e.  *Ring  /\ 
A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f
) )  /\  (
( x h x )  =  ( 0g
`  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) ) )  <->  ( F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
.,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y
) )  =  ( y  .,  x ) ) ) ) )
522, 51sbcied 3165 . . 3  |-  ( g  =  W  ->  ( [. ( Base `  g
)  /  v ]. [. ( .i `  g
)  /  h ]. [. (Scalar `  g )  /  f ]. (
f  e.  *Ring  /\  A. x  e.  v  (
( y  e.  v 
|->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f )
)  /\  ( (
x h x )  =  ( 0g `  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) ) )  <->  ( F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
.,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y
) )  =  ( y  .,  x ) ) ) ) )
53 df-phl 16820 . . 3  |-  PreHil  =  {
g  e.  LVec  |  [. ( Base `  g )  /  v ]. [. ( .i `  g )  /  h ]. [. (Scalar `  g )  /  f ]. ( f  e.  *Ring  /\ 
A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f
) )  /\  (
( x h x )  =  ( 0g
`  f )  ->  x  =  ( 0g `  g ) )  /\  A. y  e.  v  ( ( * r `  f ) `  (
x h y ) )  =  ( y h x ) ) ) }
5452, 53elrab2 3062 . 2  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  ( F  e.  *Ring  /\ 
A. x  e.  V  ( ( y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom  (ringLMod `  F
) )  /\  (
( x  .,  x
)  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x 
.,  y ) )  =  ( y  .,  x ) ) ) ) )
55 3anass 940 . 2  |-  ( ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  ( (
y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom 
(ringLMod `  F ) )  /\  ( ( x 
.,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y
) )  =  ( y  .,  x ) ) )  <->  ( W  e.  LVec  /\  ( F  e.  *Ring  /\  A. x  e.  V  ( (
y  e.  V  |->  ( y  .,  x ) )  e.  ( W LMHom 
(ringLMod `  F ) )  /\  ( ( x 
.,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y
) )  =  ( y  .,  x ) ) ) ) )
5654, 55bitr4i 244 1  |-  ( W  e.  PreHil 
<->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  (
( y  e.  V  |->  ( y  .,  x
) )  e.  ( W LMHom  (ringLMod `  F )
)  /\  ( (
x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y  .,  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924   [.wsbc 3129    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   Basecbs 13432   * rcstv 13494  Scalarcsca 13495   .icip 13497   0gc0g 13686   *Ringcsr 15895   LMHom clmhm 16058   LVecclvec 16137  ringLModcrglmod 16204   PreHilcphl 16818
This theorem is referenced by:  phllvec  16823  phlsrng  16825  phllmhm  16826  ipcj  16828  ipeq0  16832  isphld  16848  phlpropd  16849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-nul 4306
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-iota 5385  df-fv 5429  df-ov 6051  df-phl 16820
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