Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diblss Unicode version

Theorem diblss 31360
Description: The value of partial isomorphism B is a subspace of partial vector space H. TODO: use dib* specific theorems instead of dia* ones to shorten proof? (Contributed by NM, 11-Feb-2014.)
Hypotheses
Ref Expression
diblss.b  |-  B  =  ( Base `  K
)
diblss.l  |-  .<_  =  ( le `  K )
diblss.h  |-  H  =  ( LHyp `  K
)
diblss.u  |-  U  =  ( ( DVecH `  K
) `  W )
diblss.i  |-  I  =  ( ( DIsoB `  K
) `  W )
diblss.s  |-  S  =  ( LSubSp `  U )
Assertion
Ref Expression
diblss  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  e.  S )

Proof of Theorem diblss
Dummy variables  a 
b  x  h  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2284 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (Scalar `  U )  =  (Scalar `  U ) )
2 diblss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2283 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diblss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2283 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2283 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 31273 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2288 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 451 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( TEndo `  K ) `  W )  =  (
Base `  (Scalar `  U
) ) )
10 eqid 2283 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2283 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 31277 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2288 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 451 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)  =  ( Base `  U ) )
15 eqidd 2284 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2284 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( .s `  U )  =  ( .s `  U
) )
17 diblss.s . . 3  |-  S  =  ( LSubSp `  U )
1817a1i 10 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  S  =  ( LSubSp `  U
) )
19 diblss.b . . . 4  |-  B  =  ( Base `  K
)
20 diblss.l . . . 4  |-  .<_  =  ( le `  K )
21 diblss.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
2219, 20, 2, 21, 4, 11dibss 31359 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  ( Base `  U
) )
2322, 14sseqtr4d 3215 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) ) )
2419, 20, 2, 21dibn0 31343 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
25 simpll 730 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
26 simpr1 961 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  x  e.  ( ( TEndo `  K
) `  W )
)
2723adantr 451 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
I `  X )  C_  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) ) )
28 simpr2 962 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  a  e.  ( I `  X
) )
2927, 28sseldd 3181 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  a  e.  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) ) )
30 eqid 2283 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
312, 10, 3, 4, 30dvhvsca 31291 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) ) )  -> 
( x ( .s
`  U ) a )  =  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )
3225, 26, 29, 31syl12anc 1180 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x ( .s `  U ) a )  =  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
3332oveq1d 5873 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( x ( .s
`  U ) a ) ( +g  `  U
) b )  =  ( <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. ( +g  `  U ) b ) )
34 simplr 731 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( X  e.  B  /\  X  .<_  W ) )
3519, 20, 2, 10, 21dibelval1st1 31340 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)
3625, 34, 28, 35syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)
372, 10, 3tendocl 30956 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W )  /\  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( x `  ( 1st `  a
) )  e.  ( ( LTrn `  K
) `  W )
)
3825, 26, 36, 37syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x `  ( 1st `  a ) )  e.  ( ( LTrn `  K
) `  W )
)
39 eqid 2283 . . . . . . . . . 10  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
4019, 20, 2, 10, 39, 21dibelval2nd 31342 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 2nd `  a )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
4125, 34, 28, 40syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 2nd `  a )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
4219, 2, 10, 3, 39tendo0cl 30979 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )
4342ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) )  e.  ( ( TEndo `  K
) `  W )
)
4441, 43eqeltrd 2357 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 2nd `  a )  e.  ( ( TEndo `  K
) `  W )
)
452, 3tendococl 30961 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W )  /\  ( 2nd `  a )  e.  ( ( TEndo `  K
) `  W )
)  ->  ( x  o.  ( 2nd `  a
) )  e.  ( ( TEndo `  K ) `  W ) )
4625, 26, 44, 45syl3anc 1182 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x  o.  ( 2nd `  a ) )  e.  ( ( TEndo `  K
) `  W )
)
47 opelxpi 4721 . . . . . 6  |-  ( ( ( x `  ( 1st `  a ) )  e.  ( ( LTrn `  K ) `  W
)  /\  ( x  o.  ( 2nd `  a
) )  e.  ( ( TEndo `  K ) `  W ) )  ->  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >.  e.  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
4838, 46, 47syl2anc 642 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>.  e.  ( ( (
LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
49 simpr3 963 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  b  e.  ( I `  X
) )
5027, 49sseldd 3181 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  b  e.  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) ) )
51 eqid 2283 . . . . . 6  |-  ( +g  `  U )  =  ( +g  `  U )
52 eqid 2283 . . . . . 6  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
532, 10, 3, 4, 5, 51, 52dvhvadd 31282 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>.  e.  ( ( (
LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) )  /\  b  e.  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) ) )  ->  ( <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ( +g  `  U
) b )  = 
<. ( ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >. )
5425, 48, 50, 53syl12anc 1180 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. ( +g  `  U ) b )  =  <. (
( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >. )
5533, 54eqtrd 2315 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( x ( .s
`  U ) a ) ( +g  `  U
) b )  = 
<. ( ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >. )
56 fvex 5539 . . . . . . . 8  |-  ( x `
 ( 1st `  a
) )  e.  _V
57 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
58 fvex 5539 . . . . . . . . 9  |-  ( 2nd `  a )  e.  _V
5957, 58coex 5216 . . . . . . . 8  |-  ( x  o.  ( 2nd `  a
) )  e.  _V
6056, 59op1st 6128 . . . . . . 7  |-  ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x `  ( 1st `  a ) )
6160coeq1i 4843 . . . . . 6  |-  ( ( 1st `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) )  =  ( ( x `
 ( 1st `  a
) )  o.  ( 1st `  b ) )
6219, 20, 2, 10, 21dibelval1st1 31340 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 1st `  b )  e.  ( ( LTrn `  K
) `  W )
)
6325, 34, 49, 62syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 1st `  b )  e.  ( ( LTrn `  K
) `  W )
)
642, 10ltrnco 30908 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x `  ( 1st `  a ) )  e.  ( (
LTrn `  K ) `  W )  /\  ( 1st `  b )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
x `  ( 1st `  a ) )  o.  ( 1st `  b
) )  e.  ( ( LTrn `  K
) `  W )
)
6525, 38, 63, 64syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( x `  ( 1st `  a ) )  o.  ( 1st `  b
) )  e.  ( ( LTrn `  K
) `  W )
)
66 simplll 734 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  K  e.  HL )
67 hllat 29553 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
6866, 67syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  K  e.  Lat )
69 eqid 2283 . . . . . . . . . 10  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
7019, 2, 10, 69trlcl 30353 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( x `
 ( 1st `  a
) )  o.  ( 1st `  b ) )  e.  ( ( LTrn `  K ) `  W
) )  ->  (
( ( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) )  e.  B )
7125, 65, 70syl2anc 642 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) )  e.  B )
7219, 2, 10, 69trlcl 30353 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x `  ( 1st `  a ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( x `  ( 1st `  a
) ) )  e.  B )
7325, 38, 72syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) )  e.  B )
7419, 2, 10, 69trlcl 30353 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  b
)  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  e.  B )
7525, 63, 74syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( 1st `  b
) )  e.  B
)
76 eqid 2283 . . . . . . . . . 10  |-  ( join `  K )  =  (
join `  K )
7719, 76latjcl 14156 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( ( trL `  K ) `  W
) `  ( x `  ( 1st `  a
) ) )  e.  B  /\  ( ( ( trL `  K
) `  W ) `  ( 1st `  b
) )  e.  B
)  ->  ( (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) )  e.  B
)
7868, 73, 75, 77syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( ( trL `  K ) `  W
) `  ( x `  ( 1st `  a
) ) ) (
join `  K )
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) ) )  e.  B )
79 simplrl 736 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  X  e.  B )
8020, 76, 2, 10, 69trlco 30916 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x `  ( 1st `  a ) )  e.  ( (
LTrn `  K ) `  W )  /\  ( 1st `  b )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) ) 
.<_  ( ( ( ( trL `  K ) `
 W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) ) )
8125, 38, 63, 80syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) ) 
.<_  ( ( ( ( trL `  K ) `
 W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) ) )
8219, 2, 10, 69trlcl 30353 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  a
)  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  a ) )  e.  B )
8325, 36, 82syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( 1st `  a
) )  e.  B
)
8420, 2, 10, 69, 3tendotp 30950 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W )  /\  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( (
( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) )  .<_  ( (
( trL `  K
) `  W ) `  ( 1st `  a
) ) )
8525, 26, 36, 84syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) )  .<_  ( (
( trL `  K
) `  W ) `  ( 1st `  a
) ) )
86 eqid 2283 . . . . . . . . . . . . 13  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
8719, 20, 2, 86, 21dibelval1st 31339 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 1st `  a )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
8825, 34, 28, 87syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 1st `  a )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
8919, 20, 2, 10, 69, 86diatrl 31234 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  a )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  a ) )  .<_  X )
9025, 34, 88, 89syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( 1st `  a
) )  .<_  X )
9119, 20, 68, 73, 83, 79, 85, 90lattrd 14164 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) )  .<_  X )
9219, 20, 2, 86, 21dibelval1st 31339 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 1st `  b )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
9325, 34, 49, 92syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 1st `  b )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
9419, 20, 2, 10, 69, 86diatrl 31234 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  b )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  .<_  X )
9525, 34, 93, 94syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( 1st `  b
) )  .<_  X )
9619, 20, 76latjle12 14168 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( ( ( trL `  K ) `
 W ) `  ( x `  ( 1st `  a ) ) )  e.  B  /\  ( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  e.  B  /\  X  e.  B ) )  -> 
( ( ( ( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) )  .<_  X  /\  ( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  .<_  X )  <->  ( (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) )  .<_  X ) )
9768, 73, 75, 79, 96syl13anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( ( ( trL `  K ) `
 W ) `  ( x `  ( 1st `  a ) ) )  .<_  X  /\  ( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  .<_  X )  <->  ( (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) )  .<_  X ) )
9891, 95, 97mpbi2and 887 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( ( trL `  K ) `  W
) `  ( x `  ( 1st `  a
) ) ) (
join `  K )
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) ) ) 
.<_  X )
9919, 20, 68, 71, 78, 79, 81, 98lattrd 14164 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) ) 
.<_  X )
10019, 20, 2, 10, 69, 86diaelval 31223 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  <->  ( ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) )  e.  ( ( LTrn `  K
) `  W )  /\  ( ( ( trL `  K ) `  W
) `  ( (
x `  ( 1st `  a ) )  o.  ( 1st `  b
) ) )  .<_  X ) ) )
101100adantr 451 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  <->  ( ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) )  e.  ( ( LTrn `  K
) `  W )  /\  ( ( ( trL `  K ) `  W
) `  ( (
x `  ( 1st `  a ) )  o.  ( 1st `  b
) ) )  .<_  X ) ) )
10265, 99, 101mpbir2and 888 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( x `  ( 1st `  a ) )  o.  ( 1st `  b
) )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )
10361, 102syl5eqel 2367 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) )  e.  ( ( (
DIsoA `  K ) `  W ) `  X
) )
104 eqid 2283 . . . . . . . . . 10  |-  ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) )  =  ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) )
1052, 10, 3, 4, 5, 104, 52dvhfplusr 31274 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  (Scalar `  U ) )  =  ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) )
106105ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( +g  `  (Scalar `  U
) )  =  ( s  e.  ( (
TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) )
10756, 59op2nd 6129 . . . . . . . . 9  |-  ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x  o.  ( 2nd `  a
) )
10841coeq2d 4846 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x  o.  ( 2nd `  a ) )  =  ( x  o.  (
h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) ) )
10919, 2, 10, 3, 39tendo0mulr 31016 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W ) )  -> 
( x  o.  (
h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
11025, 26, 109syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x  o.  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) )  =  ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) )
111108, 110eqtrd 2315 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
x  o.  ( 2nd `  a ) )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
112107, 111syl5eq 2327 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
11319, 20, 2, 10, 39, 21dibelval2nd 31342 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 2nd `  b )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
11425, 34, 49, 113syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  ( 2nd `  b )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
115106, 112, 114oveq123d 5879 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  =  ( ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) ( s  e.  ( (
TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) ) )
116 simpllr 735 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  W  e.  H )
11719, 2, 10, 3, 39, 104tendo0pl 30980 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )  -> 
( ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) ( s  e.  ( ( TEndo `  K
) `  W ) ,  t  e.  (
( TEndo `  K ) `  W )  |->  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  ( ( s `
 h )  o.  ( t `  h
) ) ) ) ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )  =  ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) )
11866, 116, 43, 117syl21anc 1181 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) ( s  e.  ( ( TEndo `  K
) `  W ) ,  t  e.  (
( TEndo `  K ) `  W )  |->  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  ( ( s `
 h )  o.  ( t `  h
) ) ) ) ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )  =  ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) )
119115, 118eqtrd 2315 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) )
120 ovex 5883 . . . . . . 7  |-  ( ( 2nd `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  _V
121120elsnc 3663 . . . . . 6  |-  ( ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) }  <-> 
( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) )
122119, 121sylibr 203 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )
123 opelxpi 4721 . . . . 5  |-  ( ( ( ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) )  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )  ->  <. ( ( 1st `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >.  e.  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
124103, 122, 123syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  <. (
( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >.  e.  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
12519, 20, 2, 10, 39, 86, 21dibval2 31334 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
126125adantr 451 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
127124, 126eleqtrrd 2360 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  <. (
( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) ) ,  ( ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )
( +g  `  (Scalar `  U ) ) ( 2nd `  b ) ) >.  e.  (
I `  X )
)
12855, 127eqeltrd 2357 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  /\  ( x  e.  ( ( TEndo `  K
) `  W )  /\  a  e.  (
I `  X )  /\  b  e.  (
I `  X )
) )  ->  (
( x ( .s
`  U ) a ) ( +g  `  U
) b )  e.  ( I `  X
) )
1291, 9, 14, 15, 16, 18, 23, 24, 128islssd 15693 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   lecple 13215   joincjn 14078   Latclat 14151   LSubSpclss 15689   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941   DIsoAcdia 31218   DVecHcdvh 31268   DIsoBcdib 31328
This theorem is referenced by:  diblsmopel  31361  cdlemn5pre  31390  cdlemn11c  31399  dihjustlem  31406  dihord1  31408  dihord2a  31409  dihord2b  31410  dihord11c  31414  dihlsscpre  31424  dihopelvalcpre  31438  dihlss  31440  dihord6apre  31446  dihord5b  31449  dihord5apre  31452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-lss 15690  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329
  Copyright terms: Public domain W3C validator