Step |
Hyp |
Ref |
Expression |
1 |
|
paddasslem.l |
|- .<_ = ( le ` K ) |
2 |
|
paddasslem.j |
|- .\/ = ( join ` K ) |
3 |
|
paddasslem.a |
|- A = ( Atoms ` K ) |
4 |
|
paddasslem.p |
|- .+ = ( +P ` K ) |
5 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
6 |
5
|
3ad2ant1 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. Lat ) |
7 |
|
simp21 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> X C_ A ) |
8 |
|
simp1 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. HL ) |
9 |
|
simp22 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Y C_ A ) |
10 |
|
simp23 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Z C_ A ) |
11 |
3 4
|
paddssat |
|- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A ) |
12 |
8 9 10 11
|
syl3anc |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( Y .+ Z ) C_ A ) |
13 |
|
simp3l |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) ) |
14 |
1 2 3 4
|
elpaddn0 |
|- ( ( ( K e. Lat /\ X C_ A /\ ( Y .+ Z ) C_ A ) /\ ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) ) |
15 |
6 7 12 13 14
|
syl31anc |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) ) |
16 |
|
simpr |
|- ( ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( Y =/= (/) /\ Z =/= (/) ) ) |
17 |
1 2 3 4
|
paddasslem15 |
|- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) |
18 |
16 17
|
syl3anl3 |
|- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) |
19 |
18
|
3exp2 |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. A -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) ) ) |
20 |
19
|
imp |
|- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) ) |
21 |
20
|
rexlimdvv |
|- ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) |
22 |
21
|
expimpd |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) |
23 |
15 22
|
sylbid |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) |
24 |
23
|
ssrdv |
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) |