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Theorem diblss 31982
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 2297 . 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 2296 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
4 diblss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 eqid 2296 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
6 eqid 2296 . . . . 5  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
72, 3, 4, 5, 6dvhbase 31895 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  (Scalar `  U ) )  =  ( ( TEndo `  K
) `  W )
)
87eqcomd 2301 . . 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 2296 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
11 eqid 2296 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
122, 10, 3, 4, 11dvhvbase 31899 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1312eqcomd 2301 . . 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 2297 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( +g  `  U )  =  ( +g  `  U
) )
16 eqidd 2297 . 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 31981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  C_  ( Base `  U
) )
2322, 14sseqtr4d 3228 . 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 31965 . 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 3194 . . . . . 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 2296 . . . . . . 7  |-  ( .s
`  U )  =  ( .s `  U
)
312, 10, 3, 4, 30dvhvsca 31913 . . . . . 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 5889 . . . 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 31962 . . . . . . . 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 31578 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
4019, 20, 2, 10, 39, 21dibelval2nd 31964 . . . . . . . . 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 31601 . . . . . . . . 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 2370 . . . . . . 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 31583 . . . . . . 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 4737 . . . . . 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 3194 . . . . 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 2296 . . . . . 6  |-  ( +g  `  U )  =  ( +g  `  U )
52 eqid 2296 . . . . . 6  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
532, 10, 3, 4, 5, 51, 52dvhvadd 31904 . . . . 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 2328 . . 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 5555 . . . . . . . 8  |-  ( x `
 ( 1st `  a
) )  e.  _V
57 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
58 fvex 5555 . . . . . . . . 9  |-  ( 2nd `  a )  e.  _V
5957, 58coex 5232 . . . . . . . 8  |-  ( x  o.  ( 2nd `  a
) )  e.  _V
6056, 59op1st 6144 . . . . . . 7  |-  ( 1st `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x `  ( 1st `  a ) )
6160coeq1i 4859 . . . . . 6  |-  ( ( 1st `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. )  o.  ( 1st `  b ) )  =  ( ( x `
 ( 1st `  a
) )  o.  ( 1st `  b ) )
6219, 20, 2, 10, 21dibelval1st1 31962 . . . . . . . . 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 31530 . . . . . . . 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 30175 . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
7019, 2, 10, 69trlcl 30975 . . . . . . . . 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 30975 . . . . . . . . . 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 30975 . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( join `  K )  =  (
join `  K )
7719, 76latjcl 14172 . . . . . . . . 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 31538 . . . . . . . . 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 30975 . . . . . . . . . . 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 31572 . . . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
8719, 20, 2, 86, 21dibelval1st 31961 . . . . . . . . . . . 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 31856 . . . . . . . . . . 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 14180 . . . . . . . . 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 31961 . . . . . . . . . . 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 31856 . . . . . . . . . 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 14184 . . . . . . . . . 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 14180 . . . . . . 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 31845 . . . . . . . 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 2380 . . . . 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 2296 . . . . . . . . . 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 31896 . . . . . . . . 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 6145 . . . . . . . . 9  |-  ( 2nd `  <. ( x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a
) ) >. )  =  ( x  o.  ( 2nd `  a
) )
10841coeq2d 4862 . . . . . . . . . 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 31638 . . . . . . . . . . 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 2328 . . . . . . . . 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 2340 . . . . . . . 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 31964 . . . . . . . . 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 5895 . . . . . . 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 31602 . . . . . . . 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 2328 . . . . . 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 5899 . . . . . . 7  |-  ( ( 2nd `  <. (
x `  ( 1st `  a ) ) ,  ( x  o.  ( 2nd `  a ) )
>. ) ( +g  `  (Scalar `  U ) ) ( 2nd `  b ) )  e.  _V
121120elsnc 3676 . . . . . 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 4737 . . . . 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 31956 . . . . 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 2373 . . 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 2370 . 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 15709 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 1632    e. wcel 1696    C_ wss 3165   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   lecple 13231   joincjn 14094   Latclat 14167   LSubSpclss 15705   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563   DIsoAcdia 31840   DVecHcdvh 31890   DIsoBcdib 31950
This theorem is referenced by:  diblsmopel  31983  cdlemn5pre  32012  cdlemn11c  32021  dihjustlem  32028  dihord1  32030  dihord2a  32031  dihord2b  32032  dihord11c  32036  dihlsscpre  32046  dihopelvalcpre  32060  dihlss  32062  dihord6apre  32068  dihord5b  32071  dihord5apre  32074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-lss 15706  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566  df-edring 31568  df-disoa 31841  df-dvech 31891  df-dib 31951
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