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Theorem isphl 19179
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 5887 . . . . 5  |-  ( Base `  g )  e.  _V
21a1i 11 . . . 4  |-  ( g  =  W  ->  ( Base `  g )  e. 
_V )
3 fvex 5887 . . . . . 6  |-  ( .i
`  g )  e. 
_V
43a1i 11 . . . . 5  |-  ( ( g  =  W  /\  v  =  ( Base `  g ) )  -> 
( .i `  g
)  e.  _V )
5 fvex 5887 . . . . . . 7  |-  (Scalar `  g )  e.  _V
65a1i 11 . . . . . 6  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  e.  _V )
7 id 23 . . . . . . . . 9  |-  ( f  =  (Scalar `  g
)  ->  f  =  (Scalar `  g ) )
8 simpll 758 . . . . . . . . . . 11  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  g  =  W )
98fveq2d 5881 . . . . . . . . . 10  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  =  (Scalar `  W ) )
10 isphl.f . . . . . . . . . 10  |-  F  =  (Scalar `  W )
119, 10syl6eqr 2481 . . . . . . . . 9  |-  ( ( ( g  =  W  /\  v  =  (
Base `  g )
)  /\  h  =  ( .i `  g ) )  ->  (Scalar `  g
)  =  F )
127, 11sylan9eqr 2485 . . . . . . . 8  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  f  =  F )
1312eleq1d 2491 . . . . . . 7  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
f  e.  *Ring  <->  F  e.  *Ring
) )
14 simpllr 767 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  v  =  ( Base `  g
) )
15 simplll 766 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  g  =  W )
1615fveq2d 5881 . . . . . . . . . 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 2481 . . . . . . . . 9  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( Base `  g )  =  V )
1914, 18eqtrd 2463 . . . . . . . 8  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  v  =  V )
20 simplr 760 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  h  =  ( .i `  g ) )
2115fveq2d 5881 . . . . . . . . . . . . . 14  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( .i `  g )  =  ( .i `  W
) )
22 isphl.h . . . . . . . . . . . . . 14  |-  .,  =  ( .i `  W )
2321, 22syl6eqr 2481 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( .i `  g )  = 
.,  )
2420, 23eqtrd 2463 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  h  =  .,  )
2524oveqd 6318 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
y h x )  =  ( y  .,  x ) )
2619, 25mpteq12dv 4499 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
y  e.  v  |->  ( y h x ) )  =  ( y  e.  V  |->  ( y 
.,  x ) ) )
2712fveq2d 5881 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (ringLMod `  f )  =  (ringLMod `  F ) )
2815, 27oveq12d 6319 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
g LMHom  (ringLMod `  f )
)  =  ( W LMHom 
(ringLMod `  F ) ) )
2926, 28eleq12d 2504 . . . . . . . . 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 6318 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x h x )  =  ( x  .,  x ) )
3112fveq2d 5881 . . . . . . . . . . . 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 2481 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  f )  =  Z )
3430, 33eqeq12d 2444 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( x h x )  =  ( 0g
`  f )  <->  ( x  .,  x )  =  Z ) )
3515fveq2d 5881 . . . . . . . . . . . 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 2481 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  ( 0g `  g )  =  .0.  )
3837eqeq2d 2436 . . . . . . . . . 10  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x  =  ( 0g
`  g )  <->  x  =  .0.  ) )
3934, 38imbi12d 321 . . . . . . . . 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 5881 . . . . . . . . . . . . 13  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
*r `  f
)  =  ( *r `  F ) )
41 isphl.i . . . . . . . . . . . . 13  |-  .*  =  ( *r `  F )
4240, 41syl6eqr 2481 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
*r `  f
)  =  .*  )
4324oveqd 6318 . . . . . . . . . . . 12  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
x h y )  =  ( x  .,  y ) )
4442, 43fveq12d 5883 . . . . . . . . . . 11  |-  ( ( ( ( g  =  W  /\  v  =  ( Base `  g
) )  /\  h  =  ( .i `  g ) )  /\  f  =  (Scalar `  g
) )  ->  (
( *r `  f ) `  (
x h y ) )  =  (  .* 
`  ( x  .,  y ) ) )
4544, 25eqeq12d 2444 . . . . . . . . . 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 3039 . . . . . . . . 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 1335 . . . . . . . 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 3039 . . . . . . 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 715 . . . . . 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 3336 . . . . 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 3336 . . . 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 3336 . . 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 19177 . . 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 3231 . 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 986 . 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 255 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081   [.wsbc 3299    |-> cmpt 4479   ` cfv 5597  (class class class)co 6301   Basecbs 15106   *rcstv 15177  Scalarcsca 15178   .icip 15180   0gc0g 15323   *Ringcsr 18057   LMHom clmhm 18227   LVecclvec 18310  ringLModcrglmod 18377   PreHilcphl 19175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-nul 4551
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-iota 5561  df-fv 5605  df-ov 6304  df-phl 19177
This theorem is referenced by:  phllvec  19180  phlsrng  19182  phllmhm  19183  ipcj  19185  ipeq0  19189  isphld  19205  phlpropd  19206
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