Step |
Hyp |
Ref |
Expression |
1 |
|
dmplp |
|- dom +P. = ( P. X. P. ) |
2 |
|
ltrelpr |
|- |
3 |
|
0npr |
|- -. (/) e. P. |
4 |
|
ltaprlem |
|- ( C e. P. -> ( A ( C +P. A ) |
5 |
4
|
adantr |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( A ( C +P. A ) |
6 |
|
olc |
|- ( ( C +P. A ) ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) |
7 |
|
ltaprlem |
|- ( C e. P. -> ( B ( C +P. B ) |
8 |
7
|
adantr |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( B ( C +P. B ) |
9 |
|
ltsopr |
|- |
10 |
|
sotric |
|- ( ( ( B -. ( B = A \/ A |
11 |
9 10
|
mpan |
|- ( ( B e. P. /\ A e. P. ) -> ( B -. ( B = A \/ A |
12 |
11
|
adantl |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( B -. ( B = A \/ A |
13 |
|
addclpr |
|- ( ( C e. P. /\ B e. P. ) -> ( C +P. B ) e. P. ) |
14 |
|
addclpr |
|- ( ( C e. P. /\ A e. P. ) -> ( C +P. A ) e. P. ) |
15 |
13 14
|
anim12dan |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. B ) e. P. /\ ( C +P. A ) e. P. ) ) |
16 |
|
sotric |
|- ( ( ( ( C +P. B ) -. ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) |
17 |
9 15 16
|
sylancr |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. B ) -. ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) |
18 |
8 12 17
|
3imtr3d |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( -. ( B = A \/ A -. ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) |
19 |
18
|
con4d |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( ( C +P. B ) = ( C +P. A ) \/ ( C +P. A ) ( B = A \/ A |
20 |
6 19
|
syl5 |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) ( B = A \/ A |
21 |
|
df-or |
|- ( ( B = A \/ A ( -. B = A -> A |
22 |
20 21
|
syl6ib |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) ( -. B = A -> A |
23 |
22
|
com23 |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( -. B = A -> ( ( C +P. A ) A |
24 |
9 2
|
soirri |
|- -. ( C +P. A ) |
25 |
|
oveq2 |
|- ( B = A -> ( C +P. B ) = ( C +P. A ) ) |
26 |
25
|
breq2d |
|- ( B = A -> ( ( C +P. A ) ( C +P. A ) |
27 |
24 26
|
mtbiri |
|- ( B = A -> -. ( C +P. A ) |
28 |
27
|
pm2.21d |
|- ( B = A -> ( ( C +P. A ) A |
29 |
23 28
|
pm2.61d2 |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( ( C +P. A ) A |
30 |
5 29
|
impbid |
|- ( ( C e. P. /\ ( B e. P. /\ A e. P. ) ) -> ( A ( C +P. A ) |
31 |
30
|
3impb |
|- ( ( C e. P. /\ B e. P. /\ A e. P. ) -> ( A ( C +P. A ) |
32 |
31
|
3com13 |
|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( A ( C +P. A ) |
33 |
1 2 3 32
|
ndmovord |
|- ( C e. P. -> ( A ( C +P. A ) |