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Theorem dicvaddcl 31380
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 955 . . 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 2283 . . . . . . 7  |-  ( Base `  U )  =  (
Base `  U )
82, 3, 4, 5, 6, 7dicssdvh 31376 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( Base `  U ) )
9 eqid 2283 . . . . . . . . 9  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2283 . . . . . . . . 9  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
114, 9, 10, 6, 7dvhvbase 31277 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( Base `  U
)  =  ( ( ( LTrn `  K
) `  W )  X.  ( ( TEndo `  K
) `  W )
) )
1211eqcomd 2288 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
1312adantr 451 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( ( LTrn `  K ) `  W
)  X.  ( (
TEndo `  K ) `  W ) )  =  ( Base `  U
) )
148, 13sseqtr4d 3215 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  C_  ( (
( LTrn `  K ) `  W )  X.  (
( TEndo `  K ) `  W ) ) )
15143adant3 975 . . . 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 983 . . . 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 3181 . . 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 984 . . . 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 3181 . . 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 2283 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
21 dicvaddcl.p . . . 4  |-  .+  =  ( +g  `  U )
22 eqid 2283 . . . 4  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
234, 9, 10, 6, 20, 21, 22dvhvadd 31282 . . 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 1180 . 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 31379 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  X  e.  ( I `  Q
) )  ->  ( 2nd `  X )  e.  ( ( TEndo `  K
) `  W )
)
26253adant3r 1179 . . . . 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 31379 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
)
28273adant3l 1178 . . . . 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 2283 . . . . . . . 8  |-  ( oc
`  K )  =  ( oc `  K
)
302, 29, 3, 4lhpocnel 30207 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( oc
`  K ) `  W )  e.  A  /\  -.  ( ( oc
`  K ) `  W )  .<_  W ) )
31303ad2ant1 976 . . . . . 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 956 . . . . . 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 2283 . . . . . . 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 30768 . . . . . 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 1182 . . . . 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 2283 . . . . . 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 31212 . . . . 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 1182 . . . 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 31274 . . . . . . 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 976 . . . . . 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 5875 . . . . 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 5527 . . . 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 2283 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
442, 3, 4, 43, 9, 5dicelval1sta 31377 . . . . . 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 1179 . . . . 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 31377 . . . . . 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 1178 . . . . 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 4848 . . . 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 2326 . . 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 30970 . . . . 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 1182 . . . 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 2357 . . 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 5539 . . . . . 6  |-  ( 1st `  X )  e.  _V
54 fvex 5539 . . . . . 6  |-  ( 1st `  Y )  e.  _V
5553, 54coex 5216 . . . . 5  |-  ( ( 1st `  X )  o.  ( 1st `  Y
) )  e.  _V
56 ovex 5883 . . . . 5  |-  ( ( 2nd `  X ) ( +g  `  (Scalar `  U ) ) ( 2nd `  Y ) )  e.  _V
572, 3, 4, 43, 9, 10, 5, 55, 56dicopelval 31367 . . . 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 975 . . 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 888 . 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 2357 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   lecple 13215   occoc 13216   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268   DIsoCcdic 31362
This theorem is referenced by:  diclss  31383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-dvech 31269  df-dic 31363
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