Step |
Hyp |
Ref |
Expression |
1 |
|
opex |
|- <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. e. _V |
2 |
|
oveq1 |
|- ( w = A -> ( w .R u ) = ( A .R u ) ) |
3 |
|
oveq1 |
|- ( v = B -> ( v .R f ) = ( B .R f ) ) |
4 |
3
|
oveq2d |
|- ( v = B -> ( -1R .R ( v .R f ) ) = ( -1R .R ( B .R f ) ) ) |
5 |
2 4
|
oveqan12d |
|- ( ( w = A /\ v = B ) -> ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) = ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) ) |
6 |
|
oveq1 |
|- ( v = B -> ( v .R u ) = ( B .R u ) ) |
7 |
|
oveq1 |
|- ( w = A -> ( w .R f ) = ( A .R f ) ) |
8 |
6 7
|
oveqan12rd |
|- ( ( w = A /\ v = B ) -> ( ( v .R u ) +R ( w .R f ) ) = ( ( B .R u ) +R ( A .R f ) ) ) |
9 |
5 8
|
opeq12d |
|- ( ( w = A /\ v = B ) -> <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. = <. ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) , ( ( B .R u ) +R ( A .R f ) ) >. ) |
10 |
|
oveq2 |
|- ( u = C -> ( A .R u ) = ( A .R C ) ) |
11 |
|
oveq2 |
|- ( f = D -> ( B .R f ) = ( B .R D ) ) |
12 |
11
|
oveq2d |
|- ( f = D -> ( -1R .R ( B .R f ) ) = ( -1R .R ( B .R D ) ) ) |
13 |
10 12
|
oveqan12d |
|- ( ( u = C /\ f = D ) -> ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) = ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) ) |
14 |
|
oveq2 |
|- ( u = C -> ( B .R u ) = ( B .R C ) ) |
15 |
|
oveq2 |
|- ( f = D -> ( A .R f ) = ( A .R D ) ) |
16 |
14 15
|
oveqan12d |
|- ( ( u = C /\ f = D ) -> ( ( B .R u ) +R ( A .R f ) ) = ( ( B .R C ) +R ( A .R D ) ) ) |
17 |
13 16
|
opeq12d |
|- ( ( u = C /\ f = D ) -> <. ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) , ( ( B .R u ) +R ( A .R f ) ) >. = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. ) |
18 |
9 17
|
sylan9eq |
|- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. ) |
19 |
|
df-mul |
|- x. = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } |
20 |
|
df-c |
|- CC = ( R. X. R. ) |
21 |
20
|
eleq2i |
|- ( x e. CC <-> x e. ( R. X. R. ) ) |
22 |
20
|
eleq2i |
|- ( y e. CC <-> y e. ( R. X. R. ) ) |
23 |
21 22
|
anbi12i |
|- ( ( x e. CC /\ y e. CC ) <-> ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) ) |
24 |
23
|
anbi1i |
|- ( ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) <-> ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) ) |
25 |
24
|
oprabbii |
|- { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } |
26 |
19 25
|
eqtri |
|- x. = { <. <. x , y >. , z >. | ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) } |
27 |
1 18 26
|
ov3 |
|- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. x. <. C , D >. ) = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. ) |