Step |
Hyp |
Ref |
Expression |
1 |
|
padd0.a |
|- A = ( Atoms ` K ) |
2 |
|
padd0.p |
|- .+ = ( +P ` K ) |
3 |
|
simpl1 |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> K e. B ) |
4 |
|
simpl2 |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> Y C_ A ) |
5 |
|
sstr |
|- ( ( Z C_ W /\ W C_ A ) -> Z C_ A ) |
6 |
5
|
ancoms |
|- ( ( W C_ A /\ Z C_ W ) -> Z C_ A ) |
7 |
6
|
ad2ant2l |
|- ( ( ( Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> Z C_ A ) |
8 |
7
|
3adantl1 |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> Z C_ A ) |
9 |
3 4 8
|
3jca |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> ( K e. B /\ Y C_ A /\ Z C_ A ) ) |
10 |
|
simprl |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> X C_ Y ) |
11 |
1 2
|
paddss1 |
|- ( ( K e. B /\ Y C_ A /\ Z C_ A ) -> ( X C_ Y -> ( X .+ Z ) C_ ( Y .+ Z ) ) ) |
12 |
9 10 11
|
sylc |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> ( X .+ Z ) C_ ( Y .+ Z ) ) |
13 |
1 2
|
paddss2 |
|- ( ( K e. B /\ W C_ A /\ Y C_ A ) -> ( Z C_ W -> ( Y .+ Z ) C_ ( Y .+ W ) ) ) |
14 |
13
|
3com23 |
|- ( ( K e. B /\ Y C_ A /\ W C_ A ) -> ( Z C_ W -> ( Y .+ Z ) C_ ( Y .+ W ) ) ) |
15 |
14
|
imp |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ Z C_ W ) -> ( Y .+ Z ) C_ ( Y .+ W ) ) |
16 |
15
|
adantrl |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> ( Y .+ Z ) C_ ( Y .+ W ) ) |
17 |
12 16
|
sstrd |
|- ( ( ( K e. B /\ Y C_ A /\ W C_ A ) /\ ( X C_ Y /\ Z C_ W ) ) -> ( X .+ Z ) C_ ( Y .+ W ) ) |
18 |
17
|
ex |
|- ( ( K e. B /\ Y C_ A /\ W C_ A ) -> ( ( X C_ Y /\ Z C_ W ) -> ( X .+ Z ) C_ ( Y .+ W ) ) ) |