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Theorem hlhilset 31945
Description: The final Hilbert space constructed from a Hilbert lattice 
K and an arbitrary hyperplane  W in  K. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
Hypotheses
Ref Expression
hlhilset.h  |-  H  =  ( LHyp `  K
)
hlhilset.l  |-  L  =  ( (HLHil `  K
) `  W )
hlhilset.u  |-  U  =  ( ( DVecH `  K
) `  W )
hlhilset.v  |-  V  =  ( Base `  U
)
hlhilset.p  |-  .+  =  ( +g  `  U )
hlhilset.e  |-  E  =  ( ( EDRing `  K
) `  W )
hlhilset.g  |-  G  =  ( (HGMap `  K
) `  W )
hlhilset.r  |-  R  =  ( E sSet  <. (
* r `  ndx ) ,  G >. )
hlhilset.t  |-  .x.  =  ( .s `  U )
hlhilset.s  |-  S  =  ( (HDMap `  K
) `  W )
hlhilset.i  |-  .,  =  ( x  e.  V ,  y  e.  V  |->  ( ( S `  y ) `  x
) )
hlhilset.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
hlhilset  |-  ( ph  ->  L  =  ( {
<. ( Base `  ndx ) ,  V >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
Distinct variable groups:    x, y, K    ph, x, y    x, W, y
Allowed substitution hints:    .+ ( x, y)    R( x, y)    S( x, y)    .x. ( x, y)    U( x, y)    E( x, y)    G( x, y)    H( x, y)    ., ( x, y)    L( x, y)    V( x, y)

Proof of Theorem hlhilset
Dummy variables  w  k  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlhilset.l . 2  |-  L  =  ( (HLHil `  K
) `  W )
2 hlhilset.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
3 elex 2830 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  _V )
43adantr 451 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  _V )
52, 4syl 15 . . . 4  |-  ( ph  ->  K  e.  _V )
6 hlhilset.h . . . . . 6  |-  H  =  ( LHyp `  K
)
7 fvex 5577 . . . . . 6  |-  ( LHyp `  K )  e.  _V
86, 7eqeltri 2386 . . . . 5  |-  H  e. 
_V
98mptex 5787 . . . 4  |-  ( w  e.  H  |->  [_ K  /  k ]_ [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) )  e.  _V
10 nfcv 2452 . . . . 5  |-  F/_ k K
11 nfcv 2452 . . . . . 6  |-  F/_ k H
1210nfcsb1 3146 . . . . . 6  |-  F/_ k [_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )
1311, 12nfmpt 4145 . . . . 5  |-  F/_ k
( w  e.  H  |-> 
[_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) )
14 fveq2 5563 . . . . . . 7  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
1514, 6syl6eqr 2366 . . . . . 6  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
16 csbeq1a 3123 . . . . . 6  |-  ( k  =  K  ->  [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )  = 
[_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) )
1715, 16mpteq12dv 4135 . . . . 5  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) )  =  ( w  e.  H  |->  [_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) ) )
18 df-hlhil 31944 . . . . 5  |- HLHil  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) ) )
1910, 13, 17, 18fvmptf 5654 . . . 4  |-  ( ( K  e.  _V  /\  ( w  e.  H  |-> 
[_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) )  e.  _V )  -> 
(HLHil `  K )  =  ( w  e.  H  |->  [_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) ) )
205, 9, 19sylancl 643 . . 3  |-  ( ph  ->  (HLHil `  K )  =  ( w  e.  H  |->  [_ K  /  k ]_ [_ ( ( DVecH `  k ) `  w
)  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } ) ) )
215adantr 451 . . . 4  |-  ( (
ph  /\  w  =  W )  ->  K  e.  _V )
22 fvex 5577 . . . . . 6  |-  ( (
DVecH `  k ) `  w )  e.  _V
2322a1i 10 . . . . 5  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( DVecH `  k ) `  w )  e.  _V )
24 fvex 5577 . . . . . . 7  |-  ( Base `  u )  e.  _V
2524a1i 10 . . . . . 6  |-  ( ( ( ( ph  /\  w  =  W )  /\  k  =  K
)  /\  u  =  ( ( DVecH `  k
) `  w )
)  ->  ( Base `  u )  e.  _V )
26 id 19 . . . . . . . . . 10  |-  ( v  =  ( Base `  u
)  ->  v  =  ( Base `  u )
)
27 id 19 . . . . . . . . . . . . 13  |-  ( u  =  ( ( DVecH `  k ) `  w
)  ->  u  =  ( ( DVecH `  k
) `  w )
)
28 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  k  =  K )
2928fveq2d 5567 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  ( DVecH `  k )  =  ( DVecH `  K )
)
30 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  w  =  W )
3129, 30fveq12d 5569 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( DVecH `  k ) `  w )  =  ( ( DVecH `  K ) `  W ) )
32 hlhilset.u . . . . . . . . . . . . . 14  |-  U  =  ( ( DVecH `  K
) `  W )
3331, 32syl6eqr 2366 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( DVecH `  k ) `  w )  =  U )
3427, 33sylan9eqr 2370 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  w  =  W )  /\  k  =  K
)  /\  u  =  ( ( DVecH `  k
) `  w )
)  ->  u  =  U )
3534fveq2d 5567 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  w  =  W )  /\  k  =  K
)  /\  u  =  ( ( DVecH `  k
) `  w )
)  ->  ( Base `  u )  =  (
Base `  U )
)
36 hlhilset.v . . . . . . . . . . 11  |-  V  =  ( Base `  U
)
3735, 36syl6eqr 2366 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  w  =  W )  /\  k  =  K
)  /\  u  =  ( ( DVecH `  k
) `  w )
)  ->  ( Base `  u )  =  V )
3826, 37sylan9eqr 2370 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  v  =  V )
3938opeq2d 3840 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  <. ( Base `  ndx ) ,  v
>.  =  <. ( Base `  ndx ) ,  V >. )
4034adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  u  =  U )
4140fveq2d 5567 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( +g  `  u )  =  ( +g  `  U ) )
42 hlhilset.p . . . . . . . . . 10  |-  .+  =  ( +g  `  U )
4341, 42syl6eqr 2366 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( +g  `  u )  =  .+  )
4443opeq2d 3840 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  <. ( +g  ` 
ndx ) ,  ( +g  `  u )
>.  =  <. ( +g  ` 
ndx ) ,  .+  >.
)
4528fveq2d 5567 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  ( EDRing `
 k )  =  ( EDRing `  K )
)
4645, 30fveq12d 5569 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( EDRing `  k ) `  w )  =  ( ( EDRing `  K ) `  W ) )
47 hlhilset.e . . . . . . . . . . . . 13  |-  E  =  ( ( EDRing `  K
) `  W )
4846, 47syl6eqr 2366 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( EDRing `  k ) `  w )  =  E )
4928fveq2d 5567 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (HGMap `  k )  =  (HGMap `  K ) )
5049, 30fveq12d 5569 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
(HGMap `  k ) `  w )  =  ( (HGMap `  K ) `  W ) )
51 hlhilset.g . . . . . . . . . . . . . 14  |-  G  =  ( (HGMap `  K
) `  W )
5250, 51syl6eqr 2366 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
(HGMap `  k ) `  w )  =  G )
5352opeq2d 3840 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  <. (
* r `  ndx ) ,  ( (HGMap `  k ) `  w
) >.  =  <. (
* r `  ndx ) ,  G >. )
5448, 53oveq12d 5918 . . . . . . . . . . 11  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. )  =  ( E sSet  <. ( * r `
 ndx ) ,  G >. ) )
55 hlhilset.r . . . . . . . . . . 11  |-  R  =  ( E sSet  <. (
* r `  ndx ) ,  G >. )
5654, 55syl6eqr 2366 . . . . . . . . . 10  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. )  =  R )
5756opeq2d 3840 . . . . . . . . 9  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  <. (Scalar ` 
ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >.  =  <. (Scalar `  ndx ) ,  R >. )
5857ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k ) `  w ) sSet  <. (
* r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >.  =  <. (Scalar `  ndx ) ,  R >. )
5939, 44, 58tpeq123d 3755 . . . . . . 7  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  { <. ( Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( +g  `  u
) >. ,  <. (Scalar ` 
ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  =  { <. ( Base `  ndx ) ,  V >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  R >. } )
6040fveq2d 5567 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( .s `  u )  =  ( .s `  U ) )
61 hlhilset.t . . . . . . . . . 10  |-  .x.  =  ( .s `  U )
6260, 61syl6eqr 2366 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( .s `  u )  =  .x.  )
6362opeq2d 3840 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  <. ( .s
`  ndx ) ,  ( .s `  u )
>.  =  <. ( .s
`  ndx ) ,  .x.  >.
)
6428fveq2d 5567 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (HDMap `  k )  =  (HDMap `  K ) )
6564, 30fveq12d 5569 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
(HDMap `  k ) `  w )  =  ( (HDMap `  K ) `  W ) )
66 hlhilset.s . . . . . . . . . . . . . . 15  |-  S  =  ( (HDMap `  K
) `  W )
6765, 66syl6eqr 2366 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  (
(HDMap `  k ) `  w )  =  S )
6867ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( (HDMap `  k ) `  w
)  =  S )
6968fveq1d 5565 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( (
(HDMap `  k ) `  w ) `  y
)  =  ( S `
 y ) )
7069fveq1d 5565 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( (
( (HDMap `  k
) `  w ) `  y ) `  x
)  =  ( ( S `  y ) `
 x ) )
7138, 38, 70mpt2eq123dv 5952 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
)  =  ( x  e.  V ,  y  e.  V  |->  ( ( S `  y ) `
 x ) ) )
72 hlhilset.i . . . . . . . . . 10  |-  .,  =  ( x  e.  V ,  y  e.  V  |->  ( ( S `  y ) `  x
) )
7371, 72syl6eqr 2366 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
)  =  .,  )
7473opeq2d 3840 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  <. ( .i
`  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w
) `  y ) `  x ) ) >.  =  <. ( .i `  ndx ) ,  .,  >. )
7563, 74preq12d 3748 . . . . . . 7  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  { <. ( .s `  ndx ) ,  ( .s `  u
) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v 
|->  ( ( ( (HDMap `  k ) `  w
) `  y ) `  x ) ) >. }  =  { <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )
7659, 75uneq12d 3364 . . . . . 6  |-  ( ( ( ( ( ph  /\  w  =  W )  /\  k  =  K )  /\  u  =  ( ( DVecH `  k
) `  w )
)  /\  v  =  ( Base `  u )
)  ->  ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )  =  ( { <. ( Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
7725, 76csbied 3157 . . . . 5  |-  ( ( ( ( ph  /\  w  =  W )  /\  k  =  K
)  /\  u  =  ( ( DVecH `  k
) `  w )
)  ->  [_ ( Base `  u )  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( +g  `  u
) >. ,  <. (Scalar ` 
ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )  =  ( { <. ( Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
7823, 77csbied 3157 . . . 4  |-  ( ( ( ph  /\  w  =  W )  /\  k  =  K )  ->  [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )  =  ( { <. ( Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
7921, 78csbied 3157 . . 3  |-  ( (
ph  /\  w  =  W )  ->  [_ K  /  k ]_ [_ (
( DVecH `  k ) `  w )  /  u ]_ [_ ( Base `  u
)  /  v ]_ ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( +g  `  u ) >. ,  <. (Scalar `  ndx ) ,  ( ( ( EDRing `  k
) `  w ) sSet  <.
( * r `  ndx ) ,  ( (HGMap `  k ) `  w
) >. ) >. }  u.  {
<. ( .s `  ndx ) ,  ( .s `  u ) >. ,  <. ( .i `  ndx ) ,  ( x  e.  v ,  y  e.  v  |->  ( ( ( (HDMap `  k ) `  w ) `  y
) `  x )
) >. } )  =  ( { <. ( Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
802simprd 449 . . 3  |-  ( ph  ->  W  e.  H )
81 tpex 4556 . . . . 5  |-  { <. (
Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  e.  _V
82 prex 4254 . . . . 5  |-  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. }  e.  _V
8381, 82unex 4555 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  V >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  e. 
_V
8483a1i 10 . . 3  |-  ( ph  ->  ( { <. ( Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  e. 
_V )
8520, 79, 80, 84fvmptd 5644 . 2  |-  ( ph  ->  ( (HLHil `  K
) `  W )  =  ( { <. (
Base `  ndx ) ,  V >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. (Scalar ` 
ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
861, 85syl5eq 2360 1  |-  ( ph  ->  L  =  ( {
<. ( Base `  ndx ) ,  V >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   [_csb 3115    u. cun 3184   {cpr 3675   {ctp 3676   <.cop 3677    e. cmpt 4114   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   ndxcnx 13192   sSet csts 13193   Basecbs 13195   +g cplusg 13255   * rcstv 13257  Scalarcsca 13258   .scvsca 13259   .icip 13260   HLchlt 29358   LHypclh 29991   EDRingcedring 30760   DVecHcdvh 31086  HDMapchdma 31801  HGMapchg 31894  HLHilchlh 31943
This theorem is referenced by:  hlhilsca  31946  hlhilbase  31947  hlhilplus  31948  hlhilvsca  31958  hlhilip  31959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-hlhil 31944
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