Step |
Hyp |
Ref |
Expression |
1 |
|
dmaddsr |
|- dom +R = ( R. X. R. ) |
2 |
|
ltrelsr |
|- |
3 |
|
0nsr |
|- -. (/) e. R. |
4 |
|
df-nr |
|- R. = ( ( P. X. P. ) /. ~R ) |
5 |
|
oveq1 |
|- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = ( C +R [ <. x , y >. ] ~R ) ) |
6 |
|
oveq1 |
|- ( [ <. v , u >. ] ~R = C -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = ( C +R [ <. z , w >. ] ~R ) ) |
7 |
5 6
|
breq12d |
|- ( [ <. v , u >. ] ~R = C -> ( ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) . ] ~R +R [ <. z , w >. ] ~R ) <-> ( C +R [ <. x , y >. ] ~R ) . ] ~R ) ) ) |
8 |
7
|
bibi2d |
|- ( [ <. v , u >. ] ~R = C -> ( ( [ <. x , y >. ] ~R . ] ~R <-> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) . ] ~R +R [ <. z , w >. ] ~R ) ) <-> ( [ <. x , y >. ] ~R . ] ~R <-> ( C +R [ <. x , y >. ] ~R ) . ] ~R ) ) ) ) |
9 |
|
breq1 |
|- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R . ] ~R <-> A . ] ~R ) ) |
10 |
|
oveq2 |
|- ( [ <. x , y >. ] ~R = A -> ( C +R [ <. x , y >. ] ~R ) = ( C +R A ) ) |
11 |
10
|
breq1d |
|- ( [ <. x , y >. ] ~R = A -> ( ( C +R [ <. x , y >. ] ~R ) . ] ~R ) <-> ( C +R A ) . ] ~R ) ) ) |
12 |
9 11
|
bibi12d |
|- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R . ] ~R <-> ( C +R [ <. x , y >. ] ~R ) . ] ~R ) ) <-> ( A . ] ~R <-> ( C +R A ) . ] ~R ) ) ) ) |
13 |
|
breq2 |
|- ( [ <. z , w >. ] ~R = B -> ( A . ] ~R <-> A |
14 |
|
oveq2 |
|- ( [ <. z , w >. ] ~R = B -> ( C +R [ <. z , w >. ] ~R ) = ( C +R B ) ) |
15 |
14
|
breq2d |
|- ( [ <. z , w >. ] ~R = B -> ( ( C +R A ) . ] ~R ) <-> ( C +R A ) |
16 |
13 15
|
bibi12d |
|- ( [ <. z , w >. ] ~R = B -> ( ( A . ] ~R <-> ( C +R A ) . ] ~R ) ) <-> ( A ( C +R A ) |
17 |
|
addclpr |
|- ( ( v e. P. /\ u e. P. ) -> ( v +P. u ) e. P. ) |
18 |
17
|
3ad2ant1 |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( v +P. u ) e. P. ) |
19 |
|
ltapr |
|- ( ( v +P. u ) e. P. -> ( ( x +P. w ) ( ( v +P. u ) +P. ( x +P. w ) ) |
20 |
|
ltsrpr |
|- ( [ <. x , y >. ] ~R . ] ~R <-> ( x +P. w ) |
21 |
|
ltsrpr |
|- ( [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R . ] ~R <-> ( ( v +P. x ) +P. ( u +P. w ) ) |
22 |
|
vex |
|- v e. _V |
23 |
|
vex |
|- x e. _V |
24 |
|
vex |
|- u e. _V |
25 |
|
addcompr |
|- ( y +P. z ) = ( z +P. y ) |
26 |
|
addasspr |
|- ( ( y +P. z ) +P. f ) = ( y +P. ( z +P. f ) ) |
27 |
|
vex |
|- w e. _V |
28 |
22 23 24 25 26 27
|
caov4 |
|- ( ( v +P. x ) +P. ( u +P. w ) ) = ( ( v +P. u ) +P. ( x +P. w ) ) |
29 |
|
addcompr |
|- ( ( u +P. y ) +P. ( v +P. z ) ) = ( ( v +P. z ) +P. ( u +P. y ) ) |
30 |
|
vex |
|- z e. _V |
31 |
|
addcompr |
|- ( x +P. w ) = ( w +P. x ) |
32 |
|
addasspr |
|- ( ( x +P. w ) +P. f ) = ( x +P. ( w +P. f ) ) |
33 |
|
vex |
|- y e. _V |
34 |
22 30 24 31 32 33
|
caov42 |
|- ( ( v +P. z ) +P. ( u +P. y ) ) = ( ( v +P. u ) +P. ( y +P. z ) ) |
35 |
29 34
|
eqtri |
|- ( ( u +P. y ) +P. ( v +P. z ) ) = ( ( v +P. u ) +P. ( y +P. z ) ) |
36 |
28 35
|
breq12i |
|- ( ( ( v +P. x ) +P. ( u +P. w ) ) ( ( v +P. u ) +P. ( x +P. w ) ) |
37 |
21 36
|
bitri |
|- ( [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R . ] ~R <-> ( ( v +P. u ) +P. ( x +P. w ) ) |
38 |
19 20 37
|
3bitr4g |
|- ( ( v +P. u ) e. P. -> ( [ <. x , y >. ] ~R . ] ~R <-> [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R . ] ~R ) ) |
39 |
18 38
|
syl |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R . ] ~R <-> [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R . ] ~R ) ) |
40 |
|
addsrpr |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R ) |
41 |
40
|
3adant3 |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) = [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R ) |
42 |
|
addsrpr |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = [ <. ( v +P. z ) , ( u +P. w ) >. ] ~R ) |
43 |
42
|
3adant2 |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. v , u >. ] ~R +R [ <. z , w >. ] ~R ) = [ <. ( v +P. z ) , ( u +P. w ) >. ] ~R ) |
44 |
41 43
|
breq12d |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) . ] ~R +R [ <. z , w >. ] ~R ) <-> [ <. ( v +P. x ) , ( u +P. y ) >. ] ~R . ] ~R ) ) |
45 |
39 44
|
bitr4d |
|- ( ( ( v e. P. /\ u e. P. ) /\ ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R . ] ~R <-> ( [ <. v , u >. ] ~R +R [ <. x , y >. ] ~R ) . ] ~R +R [ <. z , w >. ] ~R ) ) ) |
46 |
4 8 12 16 45
|
3ecoptocl |
|- ( ( C e. R. /\ A e. R. /\ B e. R. ) -> ( A ( C +R A ) |
47 |
46
|
3coml |
|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A ( C +R A ) |
48 |
1 2 3 47
|
ndmovord |
|- ( C e. R. -> ( A ( C +R A ) |