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Theorem dicvaddcl 34557
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 983 . . 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 2441 . . . . . . 7  |-  ( Base `  U )  =  (
Base `  U )
82, 3, 4, 5, 6, 7dicssdvh 34553 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
9 eqid 2441 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2441 . . . . . . . . 9  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
114, 9, 10, 6, 7dvhvbase 34454 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1211eqcomd 2446 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1312adantr 462 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
148, 13sseqtr4d 3390 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
15143adant3 1003 . . . 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 1011 . . . 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 3354 . . 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 1012 . . . 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 3354 . . 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 2441 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
21 dicvaddcl.p . . . 4  |-  .+  =  ( +g  `  U )
22 eqid 2441 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
234, 9, 10, 6, 20, 21, 22dvhvadd 34459 . . 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 1211 . 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 34556 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  X  e.  ( I `  Q
) )  ->  ( 2nd `  X )  e.  ( ( TEndo `  K
) `  W )
)
26253adant3r 1210 . . . . 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 34556 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
)
28273adant3l 1209 . . . . 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 2441 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
302, 29, 3, 4lhpocnel 33384 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
31303ad2ant1 1004 . . . . . 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 984 . . . . . 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 2441 . . . . . . 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 33945 . . . . . 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 1213 . . . . 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 2441 . . . . . 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 34389 . . . . 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 1213 . . . 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 34451 . . . . . . 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 1004 . . . . . 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 6107 . . . . 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 5690 . . . 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 2441 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
442, 3, 4, 43, 9, 5dicelval1sta 34554 . . . . . 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 1210 . . . . 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 34554 . . . . . 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 1209 . . . . 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 5000 . . . 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 2484 . . 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 34147 . . . . 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 1213 . . . 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 2515 . . 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 5698 . . . . . 6  |-  ( 1st `  X )  e.  _V
54 fvex 5698 . . . . . 6  |-  ( 1st `  Y )  e.  _V
5553, 54coex 6528 . . . . 5  |-  ( ( 1st `  X )  o.  ( 1st `  Y
) )  e.  _V
56 ovex 6115 . . . . 5  |-  ( ( 2nd `  X ) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  _V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 34544 . . . 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 1003 . . 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 908 . 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 2515 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 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834    o. ccom 4840   ` cfv 5415   iota_crio 6048  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   lecple 14241   occoc 14242   Atomscatm 32630   HLchlt 32717   LHypclh 33350   LTrncltrn 33467   TEndoctendo 34118   DVecHcdvh 34445   DIsoCcdic 34539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-riotaBAD 32326
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-undef 6788  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-p1 15206  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864  df-lplanes 32865  df-lvols 32866  df-lines 32867  df-psubsp 32869  df-pmap 32870  df-padd 33162  df-lhyp 33354  df-laut 33355  df-ldil 33470  df-ltrn 33471  df-trl 33525  df-tendo 34121  df-edring 34123  df-dvech 34446  df-dic 34540
This theorem is referenced by:  diclss  34560
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