Step |
Hyp |
Ref |
Expression |
1 |
|
df-nr |
|- R. = ( ( P. X. P. ) /. ~R ) |
2 |
|
oveq1 |
|- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R 0R ) = ( A .R 0R ) ) |
3 |
2
|
eqeq1d |
|- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 0R ) = 0R <-> ( A .R 0R ) = 0R ) ) |
4 |
|
1pr |
|- 1P e. P. |
5 |
|
mulsrpr |
|- ( ( ( x e. P. /\ y e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R ) |
6 |
4 4 5
|
mpanr12 |
|- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R ) |
7 |
|
mulclpr |
|- ( ( x e. P. /\ 1P e. P. ) -> ( x .P. 1P ) e. P. ) |
8 |
4 7
|
mpan2 |
|- ( x e. P. -> ( x .P. 1P ) e. P. ) |
9 |
|
mulclpr |
|- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. ) |
10 |
4 9
|
mpan2 |
|- ( y e. P. -> ( y .P. 1P ) e. P. ) |
11 |
|
addclpr |
|- ( ( ( x .P. 1P ) e. P. /\ ( y .P. 1P ) e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) |
12 |
8 10 11
|
syl2an |
|- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) |
13 |
12 12
|
anim12i |
|- ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) ) |
14 |
|
eqid |
|- ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) |
15 |
|
enreceq |
|- ( ( ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> ( [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R <-> ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) = ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) +P. 1P ) ) ) |
16 |
14 15
|
mpbiri |
|- ( ( ( ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. /\ ( ( x .P. 1P ) +P. ( y .P. 1P ) ) e. P. ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R ) |
17 |
13 16
|
sylan |
|- ( ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) /\ ( 1P e. P. /\ 1P e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R ) |
18 |
4 4 17
|
mpanr12 |
|- ( ( ( x e. P. /\ y e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R ) |
19 |
18
|
anidms |
|- ( ( x e. P. /\ y e. P. ) -> [ <. ( ( x .P. 1P ) +P. ( y .P. 1P ) ) , ( ( x .P. 1P ) +P. ( y .P. 1P ) ) >. ] ~R = [ <. 1P , 1P >. ] ~R ) |
20 |
6 19
|
eqtrd |
|- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) = [ <. 1P , 1P >. ] ~R ) |
21 |
|
df-0r |
|- 0R = [ <. 1P , 1P >. ] ~R |
22 |
21
|
oveq2i |
|- ( [ <. x , y >. ] ~R .R 0R ) = ( [ <. x , y >. ] ~R .R [ <. 1P , 1P >. ] ~R ) |
23 |
20 22 21
|
3eqtr4g |
|- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 0R ) = 0R ) |
24 |
1 3 23
|
ecoptocl |
|- ( A e. R. -> ( A .R 0R ) = 0R ) |