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Theorem diblss 31968
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 2437 . 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 2436 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diblss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2436 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2436 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 31881 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2441 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( TEndo `  K
) `  W )  =  ( Base `  (Scalar `  U ) ) )
98adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( TEndo `  K ) `  W )  =  (
Base `  (Scalar `  U
) ) )
10 eqid 2436 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2436 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 31885 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2441 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1413adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
)  =  ( Base `  U ) )
15 eqidd 2437 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2437 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( .s `  U )  =  ( .s `  U
) )
17 diblss.s . . 3  |-  S  =  ( LSubSp `  U )
1817a1i 11 . 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 31967 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  ( Base `  U
) )
2322, 14sseqtr4d 3385 . 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 31951 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
25 fvex 5742 . . . . . . 7  |-  ( x `
 ( 1st `  a
) )  e.  _V
26 vex 2959 . . . . . . . 8  |-  x  e. 
_V
27 fvex 5742 . . . . . . . 8  |-  ( 2nd `  a )  e.  _V
2826, 27coex 5413 . . . . . . 7  |-  ( x  o.  ( 2nd `  a
) )  e.  _V
2925, 28op1st 6355 . . . . . 6  |-  ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x `  ( 1st `  a ) )
3029coeq1i 5032 . . . . 5  |-  ( ( 1st `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) )  =  ( ( x `
 ( 1st `  a
) )  o.  ( 1st `  b ) )
31 simpll 731 . . . . . . 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 )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simpr1 963 . . . . . . . 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.  ( ( TEndo `  K
) `  W )
)
33 simplr 732 . . . . . . . . 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  e.  B  /\  X  .<_  W ) )
34 simpr2 964 . . . . . . . . 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 )
) )  ->  a  e.  ( I `  X
) )
3519, 20, 2, 10, 21dibelval1st1 31948 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)
3631, 33, 34, 35syl3anc 1184 . . . . . . . 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 `  a )  e.  ( ( LTrn `  K
) `  W )
)
372, 10, 3tendocl 31564 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  x  e.  ( ( TEndo `  K ) `  W )  /\  ( 1st `  a )  e.  ( ( LTrn `  K
) `  W )
)  ->  ( x `  ( 1st `  a
) )  e.  ( ( LTrn `  K
) `  W )
)
3831, 32, 36, 37syl3anc 1184 . . . . . . 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 ) )  e.  ( ( LTrn `  K
) `  W )
)
39 simpr3 965 . . . . . . . 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 )
) )  ->  b  e.  ( I `  X
) )
4019, 20, 2, 10, 21dibelval1st1 31948 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 1st `  b )  e.  ( ( LTrn `  K
) `  W )
)
4131, 33, 39, 40syl3anc 1184 . . . . . . 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 `  b )  e.  ( ( LTrn `  K
) `  W )
)
422, 10ltrnco 31516 . . . . . . 7  |-  ( ( ( 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 )
)
4331, 38, 41, 42syl3anc 1184 . . . . . 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.  ( ( LTrn `  K
) `  W )
)
44 simplll 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 )
) )  ->  K  e.  HL )
45 hllat 30161 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4644, 45syl 16 . . . . . . 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 )
) )  ->  K  e.  Lat )
47 eqid 2436 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
4819, 2, 10, 47trlcl 30961 . . . . . . . 8  |-  ( ( ( 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 )
4931, 43, 48syl2anc 643 . . . . . . 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 ) ) )  e.  B )
5019, 2, 10, 47trlcl 30961 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( x `  ( 1st `  a ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( x `  ( 1st `  a
) ) )  e.  B )
5131, 38, 50syl2anc 643 . . . . . . . 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 ) ) )  e.  B )
5219, 2, 10, 47trlcl 30961 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  b
)  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  e.  B )
5331, 41, 52syl2anc 643 . . . . . . . 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 ) `  ( 1st `  b
) )  e.  B
)
54 eqid 2436 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
5519, 54latjcl 14479 . . . . . . . 8  |-  ( ( 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
)
5646, 51, 53, 55syl3anc 1184 . . . . . . 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
) ) ) (
join `  K )
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) ) )  e.  B )
57 simplrl 737 . . . . . . 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  e.  B )
5820, 54, 2, 10, 47trlco 31524 . . . . . . . 8  |-  ( ( ( 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 ) ) ) )
5931, 38, 41, 58syl3anc 1184 . . . . . . 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 ) ) ) 
.<_  ( ( ( ( trL `  K ) `
 W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) ) )
6019, 2, 10, 47trlcl 30961 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 1st `  a
)  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  a ) )  e.  B )
6131, 36, 60syl2anc 643 . . . . . . . . 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 `  a
) )  e.  B
)
6220, 2, 10, 47, 3tendotp 31558 . . . . . . . . . 10  |-  ( ( ( 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
) ) )
6331, 32, 36, 62syl3anc 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 ) ) )  .<_  ( (
( trL `  K
) `  W ) `  ( 1st `  a
) ) )
64 eqid 2436 . . . . . . . . . . . 12  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
6519, 20, 2, 64, 21dibelval1st 31947 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 1st `  a )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
6631, 33, 34, 65syl3anc 1184 . . . . . . . . . 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 `  a )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
6719, 20, 2, 10, 47, 64diatrl 31842 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  a )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  a ) )  .<_  X )
6831, 33, 66, 67syl3anc 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 ) `  ( 1st `  a
) )  .<_  X )
6919, 20, 46, 51, 61, 57, 63, 68lattrd 14487 . . . . . . . 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 ) ) )  .<_  X )
7019, 20, 2, 64, 21dibelval1st 31947 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 1st `  b )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
7131, 33, 39, 70syl3anc 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 )
) )  ->  ( 1st `  b )  e.  ( ( ( DIsoA `  K ) `  W
) `  X )
)
7219, 20, 2, 10, 47, 64diatrl 31842 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( 1st `  b )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )  -> 
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  .<_  X )
7331, 33, 71, 72syl3anc 1184 . . . . . . . 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 ) `  ( 1st `  b
) )  .<_  X )
7419, 20, 54latjle12 14491 . . . . . . . . 9  |-  ( ( 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 ) )
7546, 51, 53, 57, 74syl13anc 1186 . . . . . . . 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 ) ) )  .<_  X  /\  ( ( ( trL `  K ) `  W
) `  ( 1st `  b ) )  .<_  X )  <->  ( (
( ( trL `  K
) `  W ) `  ( x `  ( 1st `  a ) ) ) ( join `  K
) ( ( ( trL `  K ) `
 W ) `  ( 1st `  b ) ) )  .<_  X ) )
7669, 73, 75mpbi2and 888 . . . . . . 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
) ) ) (
join `  K )
( ( ( trL `  K ) `  W
) `  ( 1st `  b ) ) ) 
.<_  X )
7719, 20, 46, 49, 56, 57, 59, 76lattrd 14487 . . . . . 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 )
) )  ->  (
( ( trL `  K
) `  W ) `  ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) ) ) 
.<_  X )
7819, 20, 2, 10, 47, 64diaelval 31831 . . . . . . 7  |-  ( ( ( 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 ) ) )
7978adantr 452 . . . . . 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 )  <->  ( ( ( x `  ( 1st `  a ) )  o.  ( 1st `  b ) )  e.  ( ( LTrn `  K
) `  W )  /\  ( ( ( trL `  K ) `  W
) `  ( (
x `  ( 1st `  a ) )  o.  ( 1st `  b
) ) )  .<_  X ) ) )
8043, 77, 79mpbir2and 889 . . . . 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 ) )  o.  ( 1st `  b
) )  e.  ( ( ( DIsoA `  K
) `  W ) `  X ) )
8130, 80syl5eqel 2520 . . . 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 ) )  e.  ( ( (
DIsoA `  K ) `  W ) `  X
) )
82 eqid 2436 . . . . . . . . 9  |-  ( 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 )
) ) )
83 eqid 2436 . . . . . . . . 9  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
842, 10, 3, 4, 5, 82, 83dvhfplusr 31882 . . . . . . . 8  |-  ( ( 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 )
) ) ) )
8584ad2antrr 707 . . . . . . 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 )
) )  ->  ( +g  `  (Scalar `  U
) )  =  ( s  e.  ( (
TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) )
8625, 28op2nd 6356 . . . . . . . 8  |-  ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x  o.  ( 2nd `  a
) )
87 eqid 2436 . . . . . . . . . . . 12  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
8819, 20, 2, 10, 87, 21dibelval2nd 31950 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  a  e.  ( I `  X
) )  ->  ( 2nd `  a )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
8931, 33, 34, 88syl3anc 1184 . . . . . . . . . 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 )
) )  ->  ( 2nd `  a )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
9089coeq2d 5035 . . . . . . . . 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 ) )  =  ( x  o.  (
h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) ) )
9119, 2, 10, 3, 87tendo0mulr 31624 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
9231, 32, 91syl2anc 643 . . . . . . . . 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.  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) )  =  ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) )
9390, 92eqtrd 2468 . . . . . . . 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  o.  ( 2nd `  a ) )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
9486, 93syl5eq 2480 . . . . . . 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
) ) >. )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
9519, 20, 2, 10, 87, 21dibelval2nd 31950 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  b  e.  ( I `  X
) )  ->  ( 2nd `  b )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
9631, 33, 39, 95syl3anc 1184 . . . . . . 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 `  b )  =  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) )
9785, 94, 96oveq123d 6102 . . . . . 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 ) ) ( 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 ) ) ) )
98 simpllr 736 . . . . . . 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 )
) )  ->  W  e.  H )
9919, 2, 10, 3, 87tendo0cl 31587 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) )  e.  ( (
TEndo `  K ) `  W ) )
10099ad2antrr 707 . . . . . . 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 ) )  e.  ( ( TEndo `  K
) `  W )
)
10119, 2, 10, 3, 87, 82tendo0pl 31588 . . . . . . 7  |-  ( ( ( 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 ) ) )
10244, 98, 100, 101syl21anc 1183 . . . . . 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 )
) )  ->  (
( 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 ) ) )
10397, 102eqtrd 2468 . . . . 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 ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) ) )
104 ovex 6106 . . . . . 6  |-  ( ( 2nd `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  _V
105104elsnc 3837 . . . . 5  |-  ( ( ( 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 ) ) )
106103, 105sylibr 204 . . . 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 )
) )  ->  (
( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } )
107 opelxpi 4910 . . . 4  |-  ( ( ( ( 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 ) ) } ) )
10881, 106, 107syl2anc 643 . . 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.  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
10923adantr 452 . . . . . . 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 ) ) )
110109, 34sseldd 3349 . . . . . 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 ) ) )
111 eqid 2436 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
1122, 10, 3, 4, 111dvhvsca 31899 . . . . . 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 ) )
>. )
11331, 32, 110, 112syl12anc 1182 . . . . 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
) ) >. )
114113oveq1d 6096 . . . 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 ) )
11589, 100eqeltrd 2510 . . . . . . 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 )
)
1162, 3tendococl 31569 . . . . . . 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 ) )
11731, 32, 115, 116syl3anc 1184 . . . . . 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 )
)
118 opelxpi 4910 . . . . . 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 )
) )
11938, 117, 118syl2anc 643 . . . . 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 ) ) )
120109, 39sseldd 3349 . . . . 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 ) ) )
121 eqid 2436 . . . . . 6  |-  ( +g  `  U )  =  ( +g  `  U )
1222, 10, 3, 4, 5, 121, 83dvhvadd 31890 . . . . 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 ) ) >. )
12331, 119, 120, 122syl12anc 1182 . . . 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 ) ) >. )
124114, 123eqtrd 2468 . . 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 ) ) >. )
12519, 20, 2, 10, 87, 64, 21dibval2 31942 . . . 4  |-  ( ( ( 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 452 . . 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 )
) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
127108, 124, 1263eltr4d 2517 . 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
) )
1281, 9, 14, 15, 16, 18, 23, 24, 127islssd 16012 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3320   {csn 3814   <.cop 3817   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876    |` cres 4880    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   lecple 13536   joincjn 14401   Latclat 14474   LSubSpclss 16008   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   TEndoctendo 31549   DIsoAcdia 31826   DVecHcdvh 31876   DIsoBcdib 31936
This theorem is referenced by:  diblsmopel  31969  cdlemn5pre  31998  cdlemn11c  32007  dihjustlem  32014  dihord1  32016  dihord2a  32017  dihord2b  32018  dihord11c  32022  dihlsscpre  32032  dihopelvalcpre  32046  dihlss  32048  dihord6apre  32054  dihord5b  32057  dihord5apre  32060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-lss 16009  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-tendo 31552  df-edring 31554  df-disoa 31827  df-dvech 31877  df-dib 31937
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