Step |
Hyp |
Ref |
Expression |
1 |
|
pssss |
|- ( A C. B -> A C_ B ) |
2 |
|
pssss |
|- ( C C. D -> C C_ D ) |
3 |
|
xpss12 |
|- ( ( A C_ B /\ C C_ D ) -> ( A X. C ) C_ ( B X. D ) ) |
4 |
1 2 3
|
syl2an |
|- ( ( A C. B /\ C C. D ) -> ( A X. C ) C_ ( B X. D ) ) |
5 |
|
simpl |
|- ( ( A C. B /\ C C. D ) -> A C. B ) |
6 |
|
pssne |
|- ( A C. B -> A =/= B ) |
7 |
6
|
necomd |
|- ( A C. B -> B =/= A ) |
8 |
|
neneq |
|- ( B =/= A -> -. B = A ) |
9 |
8
|
intnanrd |
|- ( B =/= A -> -. ( B = A /\ D = C ) ) |
10 |
5 7 9
|
3syl |
|- ( ( A C. B /\ C C. D ) -> -. ( B = A /\ D = C ) ) |
11 |
|
pssn0 |
|- ( A C. B -> B =/= (/) ) |
12 |
|
pssn0 |
|- ( C C. D -> D =/= (/) ) |
13 |
|
xp11 |
|- ( ( B =/= (/) /\ D =/= (/) ) -> ( ( B X. D ) = ( A X. C ) <-> ( B = A /\ D = C ) ) ) |
14 |
11 12 13
|
syl2an |
|- ( ( A C. B /\ C C. D ) -> ( ( B X. D ) = ( A X. C ) <-> ( B = A /\ D = C ) ) ) |
15 |
10 14
|
mtbird |
|- ( ( A C. B /\ C C. D ) -> -. ( B X. D ) = ( A X. C ) ) |
16 |
|
neqne |
|- ( -. ( B X. D ) = ( A X. C ) -> ( B X. D ) =/= ( A X. C ) ) |
17 |
16
|
necomd |
|- ( -. ( B X. D ) = ( A X. C ) -> ( A X. C ) =/= ( B X. D ) ) |
18 |
15 17
|
syl |
|- ( ( A C. B /\ C C. D ) -> ( A X. C ) =/= ( B X. D ) ) |
19 |
|
df-pss |
|- ( ( A X. C ) C. ( B X. D ) <-> ( ( A X. C ) C_ ( B X. D ) /\ ( A X. C ) =/= ( B X. D ) ) ) |
20 |
4 18 19
|
sylanbrc |
|- ( ( A C. B /\ C C. D ) -> ( A X. C ) C. ( B X. D ) ) |