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Theorem dicvaddcl 34223
Description: Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
dicvaddcl.l  |-  .<_  =  ( le `  K )
dicvaddcl.a  |-  A  =  ( Atoms `  K )
dicvaddcl.h  |-  H  =  ( LHyp `  K
)
dicvaddcl.u  |-  U  =  ( ( DVecH `  K
) `  W )
dicvaddcl.i  |-  I  =  ( ( DIsoC `  K
) `  W )
dicvaddcl.p  |-  .+  =  ( +g  `  U )
Assertion
Ref Expression
dicvaddcl  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( X  .+  Y
)  e.  ( I `
 Q ) )

Proof of Theorem dicvaddcl
Dummy variables  g  h  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 999 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 dicvaddcl.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 dicvaddcl.a . . . . . . 7  |-  A  =  ( Atoms `  K )
4 dicvaddcl.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
5 dicvaddcl.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
6 dicvaddcl.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
7 eqid 2404 . . . . . . 7  |-  ( Base `  U )  =  (
Base `  U )
82, 3, 4, 5, 6, 7dicssdvh 34219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
9 eqid 2404 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2404 . . . . . . . . 9  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
114, 9, 10, 6, 7dvhvbase 34120 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1211eqcomd 2412 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1312adantr 465 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
148, 13sseqtr4d 3481 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
15143adant3 1019 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
16 simp3l 1027 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  ->  X  e.  ( I `  Q ) )
1715, 16sseldd 3445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  ->  X  e.  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
18 simp3r 1028 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  ->  Y  e.  ( I `  Q ) )
1915, 18sseldd 3445 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  ->  Y  e.  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
20 eqid 2404 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
21 dicvaddcl.p . . . 4  |-  .+  =  ( +g  `  U )
22 eqid 2404 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
234, 9, 10, 6, 20, 21, 22dvhvadd 34125 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  /\  Y  e.  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) ) )  ->  ( X  .+  Y )  =  <. ( ( 1st `  X
)  o.  ( 1st `  Y ) ) ,  ( ( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) ) >. )
241, 17, 19, 23syl12anc 1230 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( X  .+  Y
)  =  <. (
( 1st `  X
)  o.  ( 1st `  Y ) ) ,  ( ( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) ) >. )
252, 3, 4, 10, 5dicelval2nd 34222 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  X  e.  ( I `  Q
) )  ->  ( 2nd `  X )  e.  ( ( TEndo `  K
) `  W )
)
26253adant3r 1229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( 2nd `  X
)  e.  ( (
TEndo `  K ) `  W ) )
272, 3, 4, 10, 5dicelval2nd 34222 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
)
28273adant3l 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( 2nd `  Y
)  e.  ( (
TEndo `  K ) `  W ) )
29 eqid 2404 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
302, 29, 3, 4lhpocnel 33048 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
31303ad2ant1 1020 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
32 simp2 1000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
33 eqid 2404 . . . . . . 7  |-  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  =  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )
342, 3, 4, 9, 33ltrniotacl 33611 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( oc `  K ) `
 W )  e.  A  /\  -.  (
( oc `  K
) `  W )  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q )  e.  ( (
LTrn `  K ) `  W ) )
351, 31, 32, 34syl3anc 1232 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q )  e.  ( ( LTrn `  K ) `  W
) )
36 eqid 2404 . . . . . 6  |-  ( 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 )
) ) )
379, 36tendospdi2 34055 . . . . 5  |-  ( ( ( 2nd `  X
)  e.  ( (
TEndo `  K ) `  W )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )  /\  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q )  e.  ( ( LTrn `  K ) `  W
) )  ->  (
( ( 2nd `  X
) ( s  e.  ( ( TEndo `  K
) `  W ) ,  t  e.  (
( TEndo `  K ) `  W )  |->  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  ( ( s `
 h )  o.  ( t `  h
) ) ) ) ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  =  ( ( ( 2nd `  X ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  o.  ( ( 2nd `  Y
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) ) )
3826, 28, 35, 37syl3anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( ( 2nd `  X ) ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  =  ( ( ( 2nd `  X
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) )  o.  ( ( 2nd `  Y ) `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) ) )
394, 9, 10, 6, 20, 36, 22dvhfplusr 34117 . . . . . . 7  |-  ( ( 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 )
) ) ) )
40393ad2ant1 1020 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( +g  `  (Scalar `  U ) )  =  ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) )
4140oveqd 6297 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  =  ( ( 2nd `  X ) ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) ( 2nd `  Y ) ) )
4241fveq1d 5853 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( ( 2nd `  X ) ( +g  `  (Scalar `  U )
) ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  =  ( ( ( 2nd `  X ) ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) ) )
43 eqid 2404 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
442, 3, 4, 43, 9, 5dicelval1sta 34220 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  X  e.  ( I `  Q
) )  ->  ( 1st `  X )  =  ( ( 2nd `  X
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
45443adant3r 1229 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( 1st `  X
)  =  ( ( 2nd `  X ) `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
462, 3, 4, 43, 9, 5dicelval1sta 34220 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
47463adant3l 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) )
4845, 47coeq12d 4990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( 1st `  X
)  o.  ( 1st `  Y ) )  =  ( ( ( 2nd `  X ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) )  o.  ( ( 2nd `  Y
) `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  Q ) ) ) )
4938, 42, 483eqtr4rd 2456 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( 1st `  X
)  o.  ( 1st `  Y ) )  =  ( ( ( 2nd `  X ) ( +g  `  (Scalar `  U )
) ( 2nd `  Y
) ) `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  Q ) ) )
504, 9, 10, 36tendoplcl 33813 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( 2nd `  X
)  e.  ( (
TEndo `  K ) `  W )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
)  ->  ( ( 2nd `  X ) ( s  e.  ( (
TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( h  e.  ( ( LTrn `  K
) `  W )  |->  ( ( s `  h )  o.  (
t `  h )
) ) ) ( 2nd `  Y ) )  e.  ( (
TEndo `  K ) `  W ) )
511, 26, 28, 50syl3anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( 2nd `  X
) ( s  e.  ( ( TEndo `  K
) `  W ) ,  t  e.  (
( TEndo `  K ) `  W )  |->  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  ( ( s `
 h )  o.  ( t `  h
) ) ) ) ( 2nd `  Y
) )  e.  ( ( TEndo `  K ) `  W ) )
5241, 51eqeltrd 2492 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( ( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  ( (
TEndo `  K ) `  W ) )
53 fvex 5861 . . . . . 6  |-  ( 1st `  X )  e.  _V
54 fvex 5861 . . . . . 6  |-  ( 1st `  Y )  e.  _V
5553, 54coex 6738 . . . . 5  |-  ( ( 1st `  X )  o.  ( 1st `  Y
) )  e.  _V
56 ovex 6308 . . . . 5  |-  ( ( 2nd `  X ) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  _V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 34210 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. ( ( 1st `  X )  o.  ( 1st `  Y ) ) ,  ( ( 2nd `  X ) ( +g  `  (Scalar `  U )
) ( 2nd `  Y
) ) >.  e.  ( I `  Q )  <-> 
( ( ( 1st `  X )  o.  ( 1st `  Y ) )  =  ( ( ( 2nd `  X ) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  /\  (
( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  ( (
TEndo `  K ) `  W ) ) ) )
58573adant3 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( <. ( ( 1st `  X )  o.  ( 1st `  Y ) ) ,  ( ( 2nd `  X ) ( +g  `  (Scalar `  U )
) ( 2nd `  Y
) ) >.  e.  ( I `  Q )  <-> 
( ( ( 1st `  X )  o.  ( 1st `  Y ) )  =  ( ( ( 2nd `  X ) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) ) `  ( iota_ g  e.  ( ( LTrn `  K ) `  W
) ( g `  ( ( oc `  K ) `  W
) )  =  Q ) )  /\  (
( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  ( (
TEndo `  K ) `  W ) ) ) )
5949, 52, 58mpbir2and 925 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  ->  <. ( ( 1st `  X
)  o.  ( 1st `  Y ) ) ,  ( ( 2nd `  X
) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) ) >.  e.  (
I `  Q )
)
6024, 59eqeltrd 2492 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( X  e.  ( I `  Q )  /\  Y  e.  ( I `  Q
) ) )  -> 
( X  .+  Y
)  e.  ( I `
 Q ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    C_ wss 3416   <.cop 3980   class class class wbr 4397    |-> cmpt 4455    X. cxp 4823    o. ccom 4829   ` cfv 5571   iota_crio 6241  (class class class)co 6280    |-> cmpt2 6282   1stc1st 6784   2ndc2nd 6785   Basecbs 14843   +g cplusg 14911  Scalarcsca 14914   lecple 14918   occoc 14919   Atomscatm 32294   HLchlt 32381   LHypclh 33014   LTrncltrn 33131   TEndoctendo 33784   DVecHcdvh 34111   DIsoCcdic 34205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-riotaBAD 31990
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-undef 7007  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-plusg 14924  df-mulr 14925  df-sca 14927  df-vsca 14928  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190  df-tendo 33787  df-edring 33789  df-dvech 34112  df-dic 34206
This theorem is referenced by:  diclss  34226
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