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Theorem diblss 31653
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 2405 . 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 2404 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diblss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2404 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2404 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 31566 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2409 . . 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 2404 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2404 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 31570 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2409 . . 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 2405 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2405 . 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 31652 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  ( Base `  U
) )
2322, 14sseqtr4d 3345 . 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 31636 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
25 fvex 5701 . . . . . . 7  |-  ( x `
 ( 1st `  a
) )  e.  _V
26 vex 2919 . . . . . . . 8  |-  x  e. 
_V
27 fvex 5701 . . . . . . . 8  |-  ( 2nd `  a )  e.  _V
2826, 27coex 5372 . . . . . . 7  |-  ( x  o.  ( 2nd `  a
) )  e.  _V
2925, 28op1st 6314 . . . . . 6  |-  ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x `  ( 1st `  a ) )
3029coeq1i 4991 . . . . 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 31633 . . . . . . . . 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 31249 . . . . . . . 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 31633 . . . . . . . 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 31201 . . . . . . 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 29846 . . . . . . . 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 2404 . . . . . . . . 9  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
4819, 2, 10, 47trlcl 30646 . . . . . . . 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 30646 . . . . . . . . 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 30646 . . . . . . . . 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 2404 . . . . . . . . 9  |-  ( join `  K )  =  (
join `  K )
5519, 54latjcl 14434 . . . . . . . 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 31209 . . . . . . . 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 30646 . . . . . . . . . 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 31243 . . . . . . . . . 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 2404 . . . . . . . . . . . 12  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
6519, 20, 2, 64, 21dibelval1st 31632 . . . . . . . . . . 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 31527 . . . . . . . . . 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 14442 . . . . . . . 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 31632 . . . . . . . . . 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 31527 . . . . . . . . 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 14446 . . . . . . . . 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 14442 . . . . . 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 31516 . . . . . . 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 2488 . . . 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 2404 . . . . . . . . 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 2404 . . . . . . . . 9  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
842, 10, 3, 4, 5, 82, 83dvhfplusr 31567 . . . . . . . 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 6315 . . . . . . . 8  |-  ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x  o.  ( 2nd `  a
) )
87 eqid 2404 . . . . . . . . . . . 12  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
8819, 20, 2, 10, 87, 21dibelval2nd 31635 . . . . . . . . . . 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 4994 . . . . . . . . 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 31309 . . . . . . . . . 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 2436 . . . . . . . 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 2448 . . . . . . 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 31635 . . . . . . . 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 6061 . . . . . 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 31272 . . . . . . . 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 31273 . . . . . . 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 2436 . . . . 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 6065 . . . . . 6  |-  ( ( 2nd `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  _V
105104elsnc 3797 . . . . 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 4869 . . . 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 3309 . . . . . 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 2404 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
1122, 10, 3, 4, 111dvhvsca 31584 . . . . . 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 6055 . . . 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 2478 . . . . . . 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 31254 . . . . . . 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 4869 . . . . . 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 3309 . . . . 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 2404 . . . . . 6  |-  ( +g  `  U )  =  ( +g  `  U )
1222, 10, 3, 4, 5, 121, 83dvhvadd 31575 . . . . 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 2436 . . 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 31627 . . . 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 2485 . 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 15967 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 1649    e. wcel 1721    C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   lecple 13491   joincjn 14356   Latclat 14429   LSubSpclss 15963   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640   TEndoctendo 31234   DIsoAcdia 31511   DVecHcdvh 31561   DIsoBcdib 31621
This theorem is referenced by:  diblsmopel  31654  cdlemn5pre  31683  cdlemn11c  31692  dihjustlem  31699  dihord1  31701  dihord2a  31702  dihord2b  31703  dihord11c  31707  dihlsscpre  31717  dihopelvalcpre  31731  dihlss  31733  dihord6apre  31739  dihord5b  31742  dihord5apre  31745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-lss 15964  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tendo 31237  df-edring 31239  df-disoa 31512  df-dvech 31562  df-dib 31622
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